Stochastic processes are dynamic models where outcomes are uncertain and rely on probabilities. In contrast, non-stochastic processes are deterministic and predictable, with specific outcomes based on established rules.
Definition and concept of stochastic processes
Stochastic Processes: Your Guide to Randomness and Probability
Imagine you’re at a casino, watching a roulette wheel spin. The outcome of each spin is unpredictable, yet there are patterns that emerge over time. This is where stochastic processes come in. They’re like the invisible hand shaping randomness, giving us insights into the unpredictability of the world.
A stochastic process is simply a collection of random variables that change over time. Think of it like a movie, where each frame represents a random outcome and the entire film is a stochastic process. For instance, daily stock prices or temperature fluctuations over a month are prime examples of stochastic processes.
But don’t let the word “random” fool you. Stochastic processes follow certain rules, like the laws of nature. These rules allow us to make educated guesses about future events, even when we can’t predict them exactly. It’s like predicting the weather – we can’t say for sure if it will rain tomorrow, but we can use probability to estimate the chances.
So, if you’re feeling a little lost in the world of randomness, don’t worry. Stochastic processes are here to guide you, helping you make sense of the unpredictable and embrace the uncertainty that life throws your way.
Examples of common stochastic processes
Stochastic Processes and Probability: Unraveling the Randomness in the World Around Us
Hey there, data explorers! Today, we’re diving into the mind-boggling world of stochastic processes and probability theory. These concepts may sound like a mouthful, but they’re like the secret sauce that helps us make sense of the unpredictable and random events that shape our lives.
First things first, what’s a stochastic process? Think of it as a time-lapse movie of randomness. It’s a sequence of random variables, like flipping a coin over and over again. Each flip is a separate variable, but when you put them all together, you get a dynamic picture of how randomness unfolds over time.
Meet the Famous Faces of Stochastic Processes
Now, let’s introduce some of the most popular stochastic processes you may have heard of:
- Brownie on the Loose (Brownian Motion): Remember that drunk guy at the party? Brownian motion is like his path – it jumps around randomly, representing changes in temperature, stock prices, or even the number of text messages you receive in a day.
- Growing Exponentially (Geometric Brownian Motion): This one’s for the stock market enthusiasts. It’s a stochastic process that describes the random fluctuations in stock prices, where they grow exponentially with a dash of randomness.
- Phone Calls Galore (Poisson Process): Ever wondered why you get 3 missed calls in 5 minutes and then none for hours? That’s the Poisson process – a great model for random events happening at an average rate, like phone calls, accidents, or even meteorite strikes.
- Wiener’s Wild Ride (Wiener Process): This one’s a close cousin of Brownian motion, but with a twist. It’s a random walk that starts at 0 and never looks back – it just keeps wandering up and down, making it a handy mathematical tool in finance and physics.
- The Drunken Traveler (Random Walk): Imagine a drunk walking home. Their path is a random walk – a series of steps in random directions. It’s a fun way to model particle movement or even the stock market’s ups and downs.
Brownian Motion: The Dance of the Microscopic World
Imagine a microscopic particle suspended in a liquid, like a tiny boat bobbing amidst invisible waves. As it sways and dances, its path seems erratic and unpredictable. This behavior is known as Brownian motion, named after the botanist Robert Brown who first observed it in 1827.
Properties of Brownian Motion:
- Random: The particle’s movement is unpredictable, like a drunkard stumbling through a room.
- Continuous: It’s a non-stop dance, with no sudden jumps or pauses.
- Normal distribution: The particle’s displacements follow a normal distribution, forming a bell curve around an average of zero.
- Independence: The particle’s future movements are not influenced by its past path.
Applications of Brownian Motion:
This seemingly random phenomenon has profound applications in various fields:
- Financial modeling: Brownian motion is used to simulate the unpredictable fluctuations of stock prices.
- Weather prediction: It models the diffusion of pollutants and the movement of air masses.
- Biology: Brownian motion helps explain the erratic movement of cells and molecules within living organisms.
So, what’s behind this dance? Brownian motion is driven by the relentless bombardment of atoms and molecules on the microscopic particle. These tiny collisions cause the particle to bounce around, creating its characteristic random path.
In essence, Brownian motion is a testament to the hidden world of microscopic particles and the fundamental randomness that underpins our universe. It’s a reminder that even in the most seemingly chaotic systems, there’s a hidden order waiting to be discovered.
Geometric Brownian Motion: A Random Walk on the Stock Market
Picture the stock market as a whimsical wanderer, taking random steps up and down. Enter Geometric Brownian Motion—a fancy mathematical model that describes this unpredictable journey.
This model has a funny little formula: ${dS = \mu S dt + \sigma S dW_t}$, where:
– S is the stock price (the wanderer)
– $\mu$ is the average rate of change (its average step size)
– $\sigma$ is the volatility (how much it likes to wiggle)
– dW is a tiny random step (the wanderer’s whim)
Over time, our stock wanderer takes a random walk, sometimes bouncing high, sometimes taking a tumble. The key thing is that the size of its steps depends on the stock’s current price. That’s why Geometric Brownian Motion is so handy for modeling the stock market.
But don’t be fooled by the complexity of the formula. It’s just a way to describe how randomly the stock moves around. And with this model, we can make some pretty accurate predictions about the future of the stock price.
Uses of Geometric Brownian Motion
This model is like a time machine that lets us peek into the future of the stock market. Investors use it to:
– Estimate stock prices: It helps them predict the range of possible prices the stock might move in.
– Manage risk: By modeling the randomness of the market, they can make better decisions about how much risk to take.
– Develop trading strategies: With a better understanding of stock movements, traders can create more effective trading plans.
Poisson Process: Definition, characteristics, and real-world examples
Poisson Process: The Random Rainmaker
Imagine a world where events happen at a random, unpredictable pace, like raindrops falling from the sky. That’s where the Poisson Process comes in! It’s like a mathematical rainmaker that describes the occasional occurrence of events within a given time interval.
The Poisson Process is named after the brilliant mathematician Simeon Denis Poisson, who discovered that the number of events occurring in a fixed time interval follows a very specific pattern. The probability of a single event happening is directly proportional to the length of the interval. But hold your horses! This only applies to situations where events are independent of one another and don’t influence each other’s timing.
In the real world, the Poisson Process pops up in all sorts of places. From modeling customer arrivals at a store to tracking the intensity of earthquakes, this mathematical rainmaker has got it covered. It’s like a magic formula that helps us understand the randomness of life’s little surprises.
Wiener Process: The Brownian Motion’s Big Brother
Imagine a world where everything moves randomly, like a flock of birds darting through the sky. That’s the essence of a Wiener process, a stochastic process named after the legendary mathematician Norbert Wiener. It’s like Brownian motion on steroids, except it’s continuous, meaning it can move forever (or at least until something dramatic happens).
The Wiener process is a mathematical model that describes the movement of continuous-time random phenomena. Think of it as a drunken sailor meandering through the streets, taking a random step at every instant. Its path is a squiggly line, but it never doubles back on itself.
Properties of a Wiener Process:
- Independence: Each step is independent of the previous ones, so the process has no memory.
- Continuity: It moves continuously, without any sudden jumps or stops.
- Normal Distribution: Its increments (the changes in its values) follow a normal distribution.
- Mean 0: It tends to wander randomly around a fixed point.
Statistical Tests for a Wiener Process:
- Kolmogorov-Smirnov Test: Checks if the process is continuous and has no jumps.
- Mean Test: Tests whether the mean of the process is equal to a specific value.
- Variance Test: Assesses whether the variance of the process is as expected.
Understanding the Wiener process is like understanding the whims of the universe. It’s a powerful tool for modeling random phenomena, from stock market fluctuations to the motion of particles in a gas. So next time you’re wondering how the world moves, remember the Wiener process, the probabilistic puppet master pulling the strings of randomness around us.
Random Walk: A Stochastic Stroll Through Probability Land
Imagine a tipsy sailor stumbling around a ship’s deck, taking steps in a seemingly random direction. This, my friends, is the essence of a random walk, a beloved concept in the world of stochastic processes.
A random walk is just a sequence of random steps taken one after the other. Like our tipsy sailor, the path it takes is unpredictable, but it follows a fascinating set of patterns that statisticians have spent centuries studying.
There are many different types of random walks. Simple random walks, for instance, take steps of equal size and in random directions. Biased random walks, on the other hand, have a tendency to favor one direction over another. And Lévy walks are a wilder bunch, taking steps of various sizes and directions with no shame whatsoever.
These random walks have found their way into all sorts of real-world applications. They’re used to model the stock market, predict the spread of diseases, and even design algorithms that help search engines find your favorite cat videos.
So, the next time you see a tipsy sailor, don’t judge them too harshly. They might just be conducting a scientific experiment on the wonders of random walks!
Unlocking the Secrets of Stochastic Processes and Probability!
Picture this: You’re flipping a coin, not once, not twice, but a gazillion times! And guess what? Each flip is a random event. That’s where stochastic processes come in. They describe how a system evolves over time through random events.
But before we dive into the stochastic jungle, we need a quick detour: probability theory. It’s like the compass that guides us through the randomness. It tells us the chances of things happening and lays the foundation for understanding stochastic processes.
Let’s start with the basics. Probability theory is all about numbers – those elusive beasts that tell us how likely something is to happen. It’s like a game of roulette: you know that landing on red is a 50% chance, right? Probability theory helps us calculate those odds.
And it has a few golden rules, like the Law of Large Numbers. It’s like the cosmic grandma who tells us that in the long run, random events tend to average out. So, if you flip a coin a million times, you’ll probably get close to 50% heads.
Now, let’s talk about random variables. They’re like the rock stars of probability theory. They turn random events into numerical values. Imagine rolling dice – the number you land on is a random variable. And we can use probability distributions to describe the chances of getting each number.
Common distributions? We’ve got the normal distribution, which is like the bell curve you love to hate. There’s also the binomial distribution, which tells us the chances of getting a certain number of successes in a series of trials.
Now, let’s bring it all together: stochastic processes are like a bunch of random variables strung together over time. They capture how a system changes over time, and they’re used to model all sorts of real-world phenomena, from financial markets to weather patterns.
So, next time you’re flipping a coin, wondering about the weather, or trying to predict the stock market, remember that stochastic processes and probability theory are your trusty guides through the wonderland of randomness.
Random Variables: The Chameleons of Probability Theory
Picture this: you’re at a carnival, watching the roulette wheel spin. Each time the ball lands on a number, it’s like rolling a virtual dice. But instead of seeing faces with numbers, we see an outcome – a random variable.
A random variable, my friend, is a mathematical chameleon. It’s a variable that can take on different values based on some random phenomenon, like the outcome of a game of chance.
Types of Random Variables: A Smorgasbord of Shapes
The world of random variables is as diverse as a candy store. We’ve got:
- Discrete Random Variables: These variables can only take on specific, countable values. Like the number of heads when you flip a coin 10 times.
- Continuous Random Variables: These variables can take on any value within a range. Like the height of a randomly selected person.
Probability Distributions: The Blueprint of Chance
Every random variable has a blueprint that tells us how likely it is to take on different values. This blueprint is called a probability distribution.
Think of it like a weather forecast. A probability distribution tells us the chances of getting rain, sunshine, or a thunderstorm. But instead of weather patterns, we’re predicting the behavior of our random variable.
Common Probability Distributions
Just like there are different types of weather forecasts, there are different types of probability distributions:
- Normal Distribution: This bell-shaped curve is the king of distributions, describing everything from heights to test scores.
- Binomial Distribution: This distribution counts the number of successes in a series of independent trials. Like flipping a coin multiple times.
- Poisson Distribution: This distribution models the number of events occurring in a fixed interval of time. Like the number of customers arriving at a store in an hour.
Mean, Variance, and Standard Deviation: The Vital Stats
These measurements tell us a lot about our random variable:
- Mean: The average value it’s likely to take on.
- Variance: How spread out the values are. A larger variance means more variability.
- Standard Deviation: The square root of the variance. A smaller standard deviation means the values tend to cluster around the mean.
So there you have it, random variables – the building blocks of probability theory. They’re the shapeshifters that capture the uncertainty and randomness of the world around us.
Distribution: Normal distribution, binomial distribution, and other important distributions
A Journey into the Realm of Random: Unveiling the Secrets of Stochastic Processes and Probability
Prepare yourselves, dear readers, for an extraordinary adventure into the fascinating world of stochastic processes and probability theory. Together, we’ll uncover the mysteries that govern random events and peek behind the curtain of uncertainty.
Introducing Stochastic Processes: The Dance of Randomness
Imagine a world where every outcome is a roll of the dice. Stochastic processes are the mathematical tools we use to describe these captivating phenomena, revealing the patterns amidst the chaos. Just like a chess game, where each move is influenced by a myriad of factors, stochastic processes capture the intricate dance of time-dependent randomness.
Meet the Stars of Stochastic Processes
Each type of stochastic process has its own unique personality, like characters in a captivating novel. Brownian motion, for instance, dances like a drunken sailor, its path taking on an erratic, zigzagging pattern. Geometric Brownian motion adds a dash of exponential flair, describing the dance of stock prices over time.
Poisson processes are the rock stars of the random world, counting events like raindrops falling on a window pane. They’re like tiny drummers, marking the beat of time with their unpredictable tapping. And Wiener processes, the elegant cousins of Brownian motion, find their groove in financial markets, modeling the unpredictable ebbs and flows of stock prices.
Essential Probability Theory: The Guiding Light in the Storm of Randomness
Probability theory is the compass that guides us through the unpredictable seas of stochastic processes. It provides us with the tools to quantify uncertainty, predicting the likelihood of events with surprising accuracy.
We’ll delve into the realm of random variables, those enigmatic entities that can take on any numeric value. They’re like the actors in the play of randomness, each with their own unique roles and distributions. We’ll explore the iconic normal distribution, the bell curve that governs everything from human heights to IQ scores. And we’ll uncover the secrets of the binomial distribution, which models events like coin flips and website visitors.
Hypothesis Testing: Separating Truth from Fiction in Randomness
Hypothesis testing is the detective work of probability theory, helping us uncover the true nature of stochastic processes. We’ll become statistical sleuths, testing hypotheses about the mean and variance of our random variables, using tools like the t-test and chi-square test.
Beyond the Basics: Monte Carlo Simulation and Real-World Applications
Monte Carlo simulations are like virtual worlds, allowing us to explore the outcomes of complex processes without actually going through the motions. They’re the video games of probability, unlocking insights into everything from financial markets to the spread of diseases.
And let’s not forget the practical applications of stochastic processes. From modeling stock prices to predicting earthquakes, they’re the secret sauce in a vast array of scientific and technological fields. This is where the rubber meets the probability road, and we’ll explore some of the most captivating examples.
So, dear reader, prepare for a mind-bending journey into the world of stochastic processes and probability. It’s a world where randomness reigns supreme, but where the tools of probability theory illuminate the path, guiding us towards a deeper understanding of the unpredictable. Buckle up, because this is one adventure you won’t want to miss!
Get to Know the Mean, Variance, and Standard Deviation: The Trio of Statistical Superstars
Imagine you have a group of friends, and you want to get a sense of their ages. You could simply add up their ages and divide by the number of friends to get the mean, also known as the average. This gives you a single number that represents the typical age in the group.
But wait, there’s more! The variance tells you how spread out the ages are. If they’re all close to the mean, the variance will be small. But if they’re all over the place, like a pack of wild puppies, the variance will be bigger. It’s like a measure of how “clumpy” the data is.
And finally, the standard deviation is just the square root of the variance. It’s another way of expressing the spread of the data, but it has the advantage of being in the same units as the original measurements.
So, there you have it, the mean, variance, and standard deviation. They’re like the three best friends of statistics, helping us understand the central tendencies and spread of any dataset. Just think of them as the superheroes of statistical analysis, ready to conquer any numerical challenge!
Covariance: The Matchmaker of Random Variables
Imagine you have two friends, Alice and Bob. You notice that Alice is always laughing when Bob cracks jokes, and Bob always seems to be smiling when Alice is around. Could this be a coincidence, or is there a deeper connection between their personalities?
Enter covariance, the statistical detective that can uncover hidden relationships between random variables. Covariance is a measure of how much two variables co-vary, or change together.
Visualizing Covariance
Think of a scatter plot, a graph that shows the data points of two variables. If the points cluster around a straight line, the variables are said to be positively correlated. This means that when one variable goes up, the other tends to go up as well, like Alice and Bob’s laughter and smiles.
On the other hand, if the points form a V-shape, the variables are negatively correlated. When one variable increases, the other decreases, like the relationship between the price of coffee and your caffeine buzz.
Measuring Covariance
Covariance is calculated using a formula that involves the mean and standard deviation of the variables. The resulting value can be positive, negative, or zero.
- Positive covariance: The variables tend to move in the same direction.
- Negative covariance: The variables tend to move in opposite directions.
- Zero covariance: There is no linear relationship between the variables.
Interpreting Covariance
Covariance is not the same as correlation, which is a normalized measure of dependence that ranges from -1 to 1. However, covariance can give us valuable insights into the relationship between variables.
A high positive covariance suggests a strong positive relationship, while a high negative covariance indicates a strong negative relationship. A low or zero covariance means there is little or no linear relationship.
Covariance is a powerful tool for understanding the connections between random variables. It’s like the matchmaker of the statistical world, uncovering hidden relationships that can shed light on complex phenomena. So next time you’re wondering if two variables are linked, remember the power of covariance, the statistical detective that brings random variables together!
Hypothesis Testing: Definition, types of hypotheses, and statistical significance
Hypothesis Testing: The Sherlock Holmes of Stochastic Processes
Okay, let’s say you’re a private detective, just like Sherlock Holmes, and you’re on a case to figure out the truth about a stochastic process. You have a hunch, a hypothesis, that something might be going on. But how do you prove it? That’s where hypothesis testing comes in, my friend!
In the world of stochastic processes, hypothesis testing is like having a super sleuthy assistant who helps you separate the truth from the noise. You start with two possibilities: your hypothesis (Sherlock) and the alternative (Moriarty). Then, you gather evidence (data) to see if the odds favor your theory or if it’s time to pack up and go home.
The trick is to define your hypotheses carefully. You’re the grand master of your own investigation, so make sure they’re clear and statistically significant. That means your results should be unlikely to happen by chance. It’s like setting the stakes for your detective game.
There are different types of hypotheses you can test, and each one has its own set of rules. You can test the mean of your process, the variance, or even the distribution itself. It’s like using different tools in your detective tool kit to see what unlocks the truth.
So, there you have it, my fellow data detective! Hypothesis testing is your secret weapon for unraveling the mysteries of stochastic processes. Now go forth, put your deerstalker on, and solve some puzzling cases!
Testing for the Mean of a Stochastic Process: t-test and other methods
Testing the Mean of a Stochastic Process: A Statistical Adventure
Imagine you’re a detective trying to figure out the average height of a mysterious group of people. Instead of measuring each person, you decide to follow them around and observe their daily routines. This is kind of like trying to understand a stochastic process, where you’re dealing with random events that happen over time.
To test the mean of a stochastic process, we often use the trusty t-test. It’s like a statistical microscope, helping us look at a sample of our observations and make an educated guess about the mean of the whole process.
The t-test is pretty straightforward. We calculate the mean of our sample, subtract the hypothesized mean (the mean we’re trying to prove), and divide this difference by the standard deviation of our sample. The resulting number is called the t-statistic.
Then, we compare this t-statistic to a critical t-value from a table. If the t-statistic is bigger than the critical value, we can confidently reject the hypothesis that the mean of the stochastic process is equal to the hypothesized mean. It’s like finding out that the mysterious people are significantly taller (or shorter) than we thought.
Of course, there are other methods for testing the mean of a stochastic process, but they’re all similar to the t-test in their goal: to use a sample to make inferences about the entire process.
Hypothesis Testing for the Variance of a Stochastic Process
When we want to know more about the variability of a stochastic process, we can’t just roll our eyes and guess. We need to put on our statistical thinking caps and perform a hypothesis test. It’s like playing a guessing game, but with numbers and probability.
One popular test for the variance of a stochastic process is the Chi-square test. It’s like a measuring tape for variability. We take a sample from our process, calculate the variance, and then compare it to a theoretical value. If the difference is too big, we can conclude that our initial guess about the variance was wrong.
But guess what? The Chi-square test isn’t the only sheriff in town. We have other tests up our sleeves, like the likelihood ratio test. It’s like a sneaky detective, using probability to find out if our guess is spot on or way off.
Alternative Approaches
If you’re looking for a more sophisticated approach, you can try the bootstrapping method. It’s like a virtual time machine, letting us take multiple samples from our process and see how the variance changes. It’s like having a bunch of parallel universes where we can test our hypothesis over and over again.
The Moral of the Story
Don’t be afraid to get your hands dirty with hypothesis testing. It’s the key to unlocking the secrets of stochastic processes and understanding the hidden patterns in random events. So, go forth, young padawan, and test your variance with confidence!
Stochastic Processes and Probability Theory: A Guide for the Curious
Welcome to the wonderful world of randomness! In this blog post, we’re diving into the fascinating world of stochastic processes and probability theory. Prepare yourself for a wild adventure where we’ll encounter the unpredictable and try to make sense of the chaos.
What are Stochastic Processes?
Imagine tossing a coin, rolling dice, or watching the stock market fluctuate. These are all examples of stochastic processes, random events that evolve over time. They’re like mischievous kittens playing hide-and-seek, always keeping us guessing what they’ll do next.
Types of Stochastic Processes
Just like there are different types of kittens, there are different types of stochastic processes. We’ll meet the famous Brownian Motion, who dances wildly like a drunken butterfly. We’ll introduce you to the Geometric Brownian Motion, the sophisticated cousin who loves to hang out in the financial world. And let’s not forget the Poisson Process, who loves to count things like traffic accidents or radioactive decays.
Probability Theory: The Language of Randomness
To understand stochastic processes, we need to speak their language: probability theory. It’s like the Rosetta Stone for randomness, helping us decipher the patterns and make predictions about the unpredictable. We’ll explore the basics, like the rules of probability, random variables, and distributions.
Hypothesis Testing: Unmasking the Truth
Now comes the fun part: hypothesis testing! It’s like playing detective with random processes. We’ll test if the mean or variance of a stochastic process is what we expect. It’s like putting the mischievous kitten on trial and trying to prove its guilt or innocence.
Related Concepts: Where Stochastic Processes Roam
Stochastic processes don’t exist in a vacuum. They’re like mischievous kittens who love to play with their friends. We’ll meet Monte Carlo simulations, the digital twins who can help us explore complex stochastic processes. And we’ll discover the amazing ways in which these processes find applications in fields like finance, science, and even everyday life.
So, are you ready to embrace the randomness and unleash your inner detective? Join us on this stochastic adventure and let’s uncover the secrets of probability theory!
Applications of Stochastic Processes: Modeling financial markets, predicting natural phenomena, and more
Unlocking the Power of Stochastic Processes: From Market Mayhem to Predicting Nature’s Whims
Yo, probability lovers! Get ready for a wild ride into the fascinating world of stochastic processes and their mind-boggling applications. These processes are like unpredictable adventures, where random events dance and shape our world in surprising ways. Let’s dive right in!
Stochastics: The Dance of Uncertainty
Imagine a gambler rolling a dice. Every roll is a new adventure, and the outcome is uncertain. That’s the essence of a stochastic process: a series of random events that unfold over time. Examples? How about stock prices behaving like a rollercoaster or the erratic heartbeat of an EKG?
Types of Stochastic Rockstars
Brownian motion is the rebel of the bunch. It’s like a drunkard randomly stumbling about, with no clear direction. Geometric Brownian motion adds a dash of exponential flair, giving us wild swings in financial markets. The Poisson process, on the other hand, is a counting game, where events pop up at random intervals, like raindrops on a windowpane.
Probability’s Helping Hand
Probability theory is the secret weapon that makes sense of these stochastic shenanigans. It’s like the GPS that helps us navigate the maze of uncertainty. We have rules to tell us the likelihood of events, distributions to describe patterns, and measures like mean, variance, and standard deviation to understand the general vibe.
Testing the Stochastics
But sometimes, we want to know more than just the general gist. That’s where hypothesis testing comes in. We can ask questions like: “Is the average stock return significantly different from 10%?” Or “Is the variance of rainfall in the desert significantly lower than in the rainforest?” And statistical tests give us the answers we crave.
Beyond the Ivory Tower: Applications Galore
Stochastic processes aren’t just academic wonders. They’re the unsung heroes behind modeling financial markets, where they help investors navigate the treacherous waters of Wall Street. They’re also the secret sauce in predicting natural phenomena, like weather patterns and earthquake probabilities. And don’t forget Monte Carlo simulations, where we use random numbers to solve complex problems in everything from physics to engineering.
So, there you have it! Stochastic processes and probability theory: the dynamic duo that helps us understand the dance of uncertainty and uncover the hidden patterns in our unpredictable world. Embrace the randomness, my friends, and let it guide your adventures in probability land!
