MATH1180 Mathematics Assignment-UK.

Question 1: Probability of Dealing Cards
Consider cards dealt from a deck of 52 cards 1. Find the probability that a ran- domly dealt six-card hand consists of three of a kind and three of another kind. An example consists of three 10’s and three 5’s.
MATH1180 Mathematics Assignment-UK.

MATH1180 Mathematics Assignment

Question 2: Roll a Die and Toss a Coin
Roll a fair dice and toss a coin. Let X and Y be discrete random variables, with X representing the number shown on the die; and Y = 1 if the coin lands on head and Y = 0 if it lands on tail. Let Z = X + Y ; for example, if the die roll is 1 and coin lands on head, then Z(1, H) = 1 + 1 = 2.

(i) Write the sample space of the random variable Z.
(ii) Create a table for the probability mass function p(x) of Z.
(iii) Find the cumulative distribution function, F(z), for Z and plot it against the values of z.
(iv) Calculate the expectation E(Z).
(v) Calculate the variance Var(Z).

Question 3: Joint Random Variables
Let X, Y have the joint PMF as shown in the following table.

(i) Find the marginal PMFs for X and Y .
(ii) Find

(iii) Find P(Y = 1 | X = 2) and the conditional probability mass function pY |X=1 of Y given X = 1.
(iv) Find the conditional exception μY |X(x).
(v) Find Cov(X, Y ) covariance for X and Y

Question 4: Continuous Joint Distribution
Let the random variables X and Y have the joint PDF

(i) Find the constant ↵ so that f(x, y) is a joint probability mass function.
(ii) Find

(iii) Find the marginal probability density function fY (y) for Y

(iv) Find the conditional probability density function fX|Y =y(x). Hence write an integral formula for the expectation E (X | Y ). You do not need to evaluate the integral.

Question 5: Regression and Time Series
In this question you will be working with dataset longley which can be view in R by running the code data(longley). The objective is to predict the number of people employed from economic variables. You’ll need to conduct most of your analysis in R and produce screenshot of the output together with suitable comments.

The dataset has several numerical columns: GNP. deflator, GNP, Unemployed,Armed.Forces, Population, Year, Employed. The response variable is Employed.

(i) Write down the formula for correlation coecient for two random variables X and Y and use R to find pairwise correlation between the variables of longley.

(ii) Create a multiple regression model regression Employed on all available variables of the dataset. Produce a summary and ANOVA of the regression and comment on the significance of the parameter estimates.

MATH1180 Mathematics Assignment-UK.

MATH1180 Mathematics Assignment

(iii) Hence using above produce a reduced model by selecting the most relevant explanatory variables and write the equation relating Employed to the relevant explanatory variables.

(iv) State the assumptions underlying linear regression models. Produce the diagnostic plots for your reduced model found in (iii) and comment on the validity of the model.

(v) Produce a time series plot of Employed against years and produce the auto-correlation plot or values.

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