ST1111 Probability Models Assignment Sample NUIG Ireland

ST1111 Probability Models is a course offered by the Department of Statistics at the National University in Ireland, Galway. The course covers the basics of probability and statistical inference. The course aims to provide students with a strong foundation in probability theory and its application to statistical modelling. 

The first part of the course covers basic probability theory, including random variables, expected values, distributions, and limit theorems. The second part of the course introduces different types of statistical models, including linear regression, correlation analysis, k-means clustering, and maximum likelihood estimation. Each type of model is motivated by real-world examples and illustrated with worked-out problems. 

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In this section, we are describing some assigned activities. These are:

Assignment Activity 1: Model basic discrete random variables, defining the distribution function and the probability function for Uniform, Bernoulli, Binomial, Geometric, Negative Binomial, Hypergeometric, Multivariate hypergeometric, and Poisson distributions.

There are many different types of discrete random variables, each with its corresponding probability function. The most common ones are the Uniform, Bernoulli, Binomial, Geometric, Negative Binomial, Hypergeometric, Multivariate hypergeometric, and Poisson distributions.

The Uniform Distribution is perhaps the simplest of all the discrete distributions. It is defined by a single parameter, the lower limit a and the upper limit b. The probability function is: Pr(X=x) = 1/(b-a), for all x such that a ≤ x ≤ b. That is, every value within the specified range is equally likely to occur.

The Bernoulli distribution is used to model dichotomous outcomes (i.e., outcomes that can only have two possible values, such as success/failure, heads/tails, etc.). It is defined by a single parameter, the probability p of success. The probability function is: Pr(X=x) = p^x (1-p)^1-x , for x = 0, 1. That is, the only possible values for X are 0 (failure) and 1 (success), and the probability of success is p.

The Binomial distribution is used to model the number of successes in a fixed number n of trials, where each trial has only two possible outcomes (i.e., it is a Bernoulli experiment). It is defined by two parameters, the probability p of success and the number n of trials. The probability function is: Pr(X=x) = C(n,x) p^x (1-p)^n-x , for x = 0, 1, …, n. Here, C(n,x) is the number of ways of choosing x successes out of n trials.

The Geometric distribution is used to model the number of trials X required to get the first success, where each trial has only two possible outcomes (again, a Bernoulli experiment). It is defined by a single parameter, the probability p of success. The probability function is: Pr(X=x) = (1-p)^x-1 p , for x = 1, 2, …. That is, the only possible values for X are the positive integers, and the probability of success is p.

The Negative Binomial distribution is used to model the number of trials X required to get the rth success, where each trial has only two possible outcomes (again, a Bernoulli experiment). It is defined by two parameters, the probability p of success and the number r of successes. The probability function is: Pr(X=x) = C(x-1,r-1) p^r (1-p)^x-r , for x = r, r+1, …. That is, the only possible values for X are the positive integers ≥ r, and the probability of success is p.

The Hypergeometric distribution is used to model the number of successes in a fixed number n of draws from a population of size N, without replacement, where the probability of success varies from draw to draw. It is defined by three parameters, the population size N, the number n of draws, and the success probability p. The probability function is: Pr(X=x) = C(N-n,x) C(n,N-x) p^x (1-p)^n-x , for x = 0, 1, …, n. Note that this is different from the Binomial distribution, which models the number of successes in a fixed number n of draws from a population with a constant success probability p.

The Multivariate Hypergeometric distribution is used to model the number of successes in a fixed number n of draws from a population of size N, without replacement, where the probability of success varies from draw to draw and there are more than two possible outcomes. It is defined by three parameters, the population size N, the number n of draws, and the success probabilities p1, p2, …, pk. The probability function is: Pr(X1=x1, X2=x2, …, Xk=xk) = C(N-n,x1) C(n-x1,x2) C(n-x1-x2,x3) … p1^x1 p2^x2 … pk^mk , for x1 + x2 + … + xk = n. Note that this is different from the Multinomial distribution, which models the number of successes in a fixed number n of draws from a population with constant success probabilities p1, p2, …, pk.

The Poisson distribution is used to model the number of events X that occur in a given time interval, where the event rate is constant. It is defined by a single parameter, the event rate lambda. The probability function is: Pr(X=x) = lambda^x e^-lambda / x! , for x = 0, 1, 2, …. That is, the only possible values for X are the non-negative integers, and the event rate is lambda.

Assignment Activity 2: Demonstrate the idea underlying the probability density function and the distribution function of a continuous random variable and be able to define these for continuous Uniform, Normal, Gamma, Exponential, Chi-square, t, F, Beta, and Lognormal distributions.

A probability density function (PDF) is a mathematical function that describes the shape of a continuous random variable. It tells you the probability of finding the variable somewhere within a given range. In other words, it tells you how likely it is to roll a certain number on a die or to draw a certain card from a deck.

The distribution function (DF) of a continuous random variable is simply the integral of the PDF over all space. It tells you the probability that the random variable will take on any particular value between two points. In other words, it tells you what percentage of time you would get a result between two numbers if you rolled a die or drew cards from a deck an infinite number of times.

The Uniform distribution is a continuous random variable with a PDF that is constant over a given range. That is, the probability of finding the variable anywhere within that range is the same. The DF of a Uniform distribution is simply the integral of the PDF over all space. It tells you the probability that the random variable will take on any particular value between two points.

The Normal distribution is a continuous random variable with a PDF that is bell-shaped. That is, it has a peak in the middle and tails that extend to infinity. The DF of a Normal distribution is simply the integral of the PDF over all space. It tells you the probability that the random variable will take on any particular value between two points.

The Gamma distribution is a continuous random variable with a PDF that is shaped like a pyramid. That is, it has a peak in the middle and tails that extend to infinity. The DF of a Gamma distribution is simply the integral of the PDF over all space. It tells you the probability that the random variable will take on any particular value between two points.

The Exponential distribution is a continuous random variable with a PDF that is shaped like an exponential curve. That is, it has a peak in the middle and tails that extend to infinity. The DF of an Exponential distribution is simply the integral of the PDF over all space. It tells you the probability that the random variable will take on any particular value between two points.

The Chi-square distribution is a continuous random variable with a PDF that is shaped like a chi-square curve. That is, it has a peak in the middle and tails that extend to infinity. The DF of a Chi-square distribution is simply the integral of the PDF over all space. It tells you the probability that the random variable will take on any particular value between two points.

The t distribution is a continuous random variable with a PDF that is shaped like a t-shaped curve. That is, it has a peak in the middle and tails that extend to infinity. The DF of a t distribution is simply the integral of the PDF over all space. It tells you the probability that the random variable will take on any particular value between two points.

The F distribution is a continuous random variable with a PDF that is shaped like an F-shaped curve. That is, it has a peak in the middle and tails that extend to infinity. The DF of an F distribution is simply the integral of the PDF over all space. It tells you the probability that the random variable will take on any particular value between two points.

The Beta distribution is a continuous random variable with a PDF that is shaped like a beta curve. That is, it has a peak in the middle and tails that extend to infinity. The DF of a Beta distribution is simply the integral of the PDF over all space. It tells you the probability that the random variable will take on any particular value between two points.

The Lognormal distribution is a continuous random variable with a PDF that is shaped like a lognormal curve. That is, it has a peak in the middle and tails that extend to infinity. The DF of a Lognormal distribution is simply the integral of the PDF over all space. It tells you the probability that the random variable will take on any particular value between two points.

Assignment Activity 3: Evaluate probabilities and quantiles associated with probability distributions, by calculation, by reference to statistical tables, and by use of software, R statistical programming, and Minitab.

There are three main ways to evaluate probabilities and quantiles associated with probability distributions: by calculation, by reference to statistical tables, and by use of the software.

The calculation is the most straightforward method, but it can be time-consuming and tedious if you’re dealing with a lot of data. Reference to statistical tables is a more efficient way to go about it if you’re working with well-known distributions. Finally, using software like R is the best option if you want to automate the process or are working with complex distributions.

When it comes to calculating probabilities and quantiles, there are two main methods: the algebraic method and the numerical method.

The algebraic method is the more straightforward of the two, as it simply involves plugging values into a formula. However, it can be time-consuming if you’re working with a lot of data.

The numerical method is more efficient, as it allows you to calculate probabilities and quantiles without having to first find the PDF or CDF. You can simply use software like R to do this for you.

Finally, if you want to automate the process or are working with complex distributions, using software like R is the best option. This will allow you to calculate probabilities and quantiles without having to first find the PDF or CDF. Plus, it’s a good way to check your work if you’re using the algebraic method.

Assignment Activity 4: Demonstrate the importance of the first two moments of discrete and continuous random variables as summary measures of distribution, and be able to compute the mean and variance of certain discrete variables.

There are a couple of reasons why the first two moments are important summary measures of distribution. First, they give us a way to measure the central tendency of the data. Second, they can be used to compute other important measures, like the variance and standard deviation.

The first moment is simply the mean, which gives us a sense of where the centre of our distribution is. The second moment is the variance, which tells us how spread out our data is. Together, these two measures can give us a pretty good idea of what our data looks like.

While other summary measures can be useful in certain cases, the first two moments are usually going to be enough to get a good understanding of most distributions.

To compute the mean and variance of a discrete random variable, we simply need to sum up all of the values and divide them by the number of values. For a continuous random variable, we need to integrate over the entire range of values.

Assignment Activity 5: Define and derive probability generating functions, moment generating functions, and cumulant generating functions.

Probability generating functions (PGF) are mathematical functions that allow us to calculate a certain probability of an event occurring, given the PGF of the underlying distribution. 

Moment generating functions (MGM) are mathematical functions that allow us to calculate all the moments of a given distribution, given the MGM of the underlying distribution. 

Cumulant generating functions (GCF) are mathematical functions that allow us to calculate all the cumulants of a given distribution, given the GCF of the underlying distribution.

All three of these functions are defined in terms of the underlying distribution’s probability mass function (PMF) or probability density function (PDF).

Assignment Activity 6: Identify the applications for which a probability generating function, a moment generating function, a cumulant generating function, and cumulants are used, and the reasons why they are used.

A probability generating function (PGF), cumulant generating function (GCF), and cumulants are used in many different ways, depending on the problem at hand. Some of the most common applications are:

1. To calculate the probability of a certain event or set of events occurring: This is done by taking the derivative of the PGF or GCF concerning time, and then integrating over all possible times that the events could have occurred.
2. To calculate a distribution’s population characteristics: This can be done by taking the derivative of the GCF concerning k (the number of different values that a random variable can take on) and then integrating it over all possible k’s.
3. To calculate the moments of distribution: This can be done by taking the derivative of the MGM concerning time and then evaluating it at t=0.
4. To calculate the cumulants of distribution: This can be done by taking the derivative of the GCF concerning k and then evaluating it at k=0.

Cumulants are used in many different fields, including physics, economics, and statistics. They are often used to characterize the distribution of a certain random variable. For example, the first cumulant is simply the mean of the distribution, while the second cumulant is the variance. Higher-order cumulants can give us information about skewness and kurtosis.

Assignment Activity 7: State the Central Limit Theorem for a sequence of independent, identically distributed random variables. Generate simulated samples from a given distribution, using statistical software such as R programming and Minitab, and compare the sampling distribution with the Normal Distribution.

The Central Limit Theorem states that the sum of a sequence of independent, identically distributed random variables will be approximately normally distributed. This theorem is important because it allows us to make inferences about the properties of a population-based on a sample. In other words, if we know the distribution of a population, we can use samples from that population to infer information about the population as a whole.

To generate a simulated sample from a given distribution, we can use statistical software such as R programming or Minitab. We can then compare the sampling distribution with the Normal Distribution to see how well the Central Limit Theorem works.

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