ST2003 Random Variables Assignment Sample NUI Galway Ireland
ST2003 Random Variables is a module that covers the basics of probability theory and its applications to random phenomena. The objectives of this course are to give students a sound understanding of probability theory, to acquaint them with the use of computers for solving problems in probability, and to introduce them to some important applications of probability theory.
Topics that will be covered in this module include discrete and continuous distributions, joint distributions, conditional distributions, expectations, moments, central limit theorem, and generating functions. Throughout the module, students will be expected to use a computer algebra system to help them solve problems.
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In this section, we are describing some assigned activities. These are:
Assignment Activity 1: Explain what is meant by jointly distributed random variables, marginal distributions, and conditional distributions.
Jointly distributed random variables are two or more random variables that are related in some way. For example, the height and weight of people are jointly distributed because taller people tend to be heavier than shorter people, on average.
Marginal distributions are the distributions of a single random variable when you ignore the other random variables. So the marginal distribution of height would be the distribution of height for a population of people, regardless of their weight. And the marginal distribution of weight would be the distribution of weight for a population of people, regardless of their height.
Conditional distributions are the distributions of a single random variable given certain information about other random variables. So the conditional distribution of height given weight would be the distribution of height for a population of people, given that we know their weight. Similarly, the conditional distribution of weight given height would be the distribution of weight for a population of people, given that we know their height.
Assignment Activity 2: Define the probability function/density function of a marginal distribution and a conditional distribution.
The probability function of a marginal distribution is simply the sum of the probabilities of all the possible events that could occur. For example, if we have a bag containing five balls – three red balls and two blue balls – then the probability function of the marginal distribution would be P(red) = 3/5 and P(blue) = 2/5.
On the other hand, the probability density function of a conditional distribution is a bit more complicated. It is a mathematical function that describes how likely it is for an event to occur, given that another event has already occurred. For example, if we know that two balls have been drawn from the bag – one red and one blue – then the probability density function of the conditional distribution would be: P(red|blue) = 1/3 and P(blue|red) = 2/3.
Assignment Activity 3: Specify the conditions under which random variables are independent.
For two random variables to be independent, means that knowing the value of one doesn’t tell you anything about the other. The only condition under which two random variables are always independent is when they are both constants (e.g., x = 2 and y = 2). In all other cases, the variables will be dependent unless one or more of the following conditions hold:
The variables are uncorrelated. This means that knowing the value of one doesn’t help you predict the value of the other, but they can have a correlation coefficient of zero.
The events associated with each variable are mutually exclusive. This means that only one event can happen at a time (e.g., flipping a coin and getting heads or tails).
The variables are independent and given a third variable. This means that the value of one doesn’t tell you anything about the other, but knowing the value of the third variable does.
Assignment Activity 4: Derive the mean and variance of linear combinations of random variables.
The mean of a linear combination of random variables is simply the sum of the means of the individual random variables. So, if X and Y are two random variables with means μX and μY, then the mean of their sum is just μX+μY.
The variance of a linear combination of random variables is more interesting. If X and Y are two random variables with variances σ2X and σ2Y, then the variance of their sum is:
Here Cov(X, Y) denotes the covariance between X and Y. So, to find the variance of a linear combination of random variables, we need to know the covariances between all the individual random variables.
Assignment Activity 5: Describe properties of sequences of random variables and Markov chains.
A sequence of random variables is a stochastic process whose realizations are sequences of random numbers. Many times these random variables are assumed to be independent and identically distributed (i.i.d.), but this is not always the case. Sequences of random variables can exhibit many interesting properties, such as stationarity, ergodicity, and mixing.
Markov chains are a type of sequence of random variables that satisfies the Markov property. This means that each element in the sequence is only dependent on the previous element in the sequence (not on any other element). Markov chains can be either finite or infinite; however, the most interesting examples are infinite since they can model long-term behaviour or even behaviour that goes on forever. Markov chains can exhibit many interesting properties, such as irreducibility, aperiodicity, and stationarity.
Assignment Activity 6: Explain what is meant by a prior distribution, a posterior distribution, and a conjugate prior distribution.
A prior distribution is a probability distribution that expresses our beliefs about a parameter before we observe any data. For example, if we believe that the chance of rain tomorrow is 50%, then our prior distribution would be a uniform distribution centred around 50%.
A posterior distribution is a probability distribution that updates our beliefs about a parameter after we observe data. For example, if we observe that it rained today, then our posterior distribution for the chance of rain tomorrow would be shifted to the right (towards 100%).
A conjugate prior is a specific type of prior distribution that “conjugates” with a certain likelihood function. This means that if we use a conjugate prior with the binomial likelihood function (which is used for modelling Bernoulli trials), then the posterior distribution will also be binomial. This is very convenient because it means that we can update our beliefs about the parameters without having to do any math beyond adding and multiplying probabilities.
Assignment Activity 7: Derive the posterior distribution for a parameter in simple cases.
In simple cases, the posterior distribution is usually a Gaussian distribution. In other words, given the observed data and the parameters of the model, it’s usually possible to calculate a “bell curve” that represents all of the possible outcomes for the parameter in question. This bell curve will be centred around the most likely value for the parameter and will become narrower as you move away from this most likely value.
This Gaussian distribution can be used to calculate different quantities such as 95% confidence intervals or predicted values. It can also be used to generate graphs that show how likely each outcome is given the data that’s been observed.
Assignment Activity 8: Demonstration of techniques with statistical software R programming.
There are several different ways to demonstrate techniques with statistical software R programming. Some of the most common methods include creating graphs, conducting simulations, and writing code.
Creating graphs is often one of the most effective ways to visualize data and results. In R programming, you can create different types of graphs including scatterplots, histograms, line graphs, and bar charts. Additionally, you can customize your graphs by adjusting colours, fonts, and other elements.
Simulations are another great way to demonstrate techniques with R programming. By running simulations, you can not only see how different methods work but also how they perform under different conditions. Additionally, simulations can help you explore the sensitivity of your results to different parameters.
Writing code is also a great way to demonstrate techniques with R programming. By writing code, you can not only show how different methods work but also how to apply them to new data. Additionally, writing code can help you automate tasks and make your analyses more reproducible.
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