Practice section 3 of 7

Calculate the mean and variance of $Y=2X+3$, where $X$ has distribution

The discrete random variable $X$ has distribution

(i) Evaluate $p$. (ii) Find $E(X)$ and $\operatorname{Var}(X)$. (iii) Write down $E(2X-3)$ and $\operatorname{Var}(2X-3)$. (iv) For two independent observations $X_1,X_2$ of $X$, find $E(X_1+X_2)$ and, using $p$ from (i), find $P(X_1+X_2=2)$.

A fair die is thrown once and $X$ is its score. Calculate $E(X)$ and show that $\operatorname{Var}(X)=35/12$. The die is thrown twice, with scores $X_1,X_2$. Tabulate the distribution of $Y=\lvert X_1-X_2\rvert$ and find $E(Y)$. Let $Z=X_1-X_2$. State with reasons, without necessarily evaluating the quantities, whether (a) $E(Z^2)=E(Y^2)$ and (b) $\operatorname{Var}(Z)=\operatorname{Var}(Y)$.

John has four keys, only one of which unlocks a door. He tries the keys randomly without replacement. X is the number of tries needed to unlock the door.

(i) Show that $P(X=3)=1/4$.

(ii) Tabulate the distribution of X.

(iii) Write down $E(X)$ and $\operatorname{Var}(X)$.

(iv) Find $P(X^2<8)$.

Alfred and Bertie each start with £100. Two fair dice are thrown. If the total is at least 5, Alfred pays £x to Bertie, where $0<x\le8$. If the total is at most 4, Bertie pays £$(x+8)$ to Alfred. Show that Alfred’s expected cash after the first game is £$(304-2x)/3$.

Find his expected cash after six games. Find x for a fair game, defined here by equal expected winnings for Alfred and Bertie.

Given $x=3$, find the variance of Alfred’s cash after the first game.

An unbiased disc has a single dot marked on one side and two dots marked on the other side. A tetrahedral die has faces marked with score of 1, 2, 3, and 4. The probability of getting a score of 1, 2, 3, and 4 is $\frac15,p,\frac15$ and $q$ respectively, where $p,q\in[0,1]$.

A game is played by throwing the disc and the die together. The random variable $S$ is the sum of the score showing on the die and twice the number of dots showing on the disc.

(i) Find $P(S=6)$.

Given that $P(S=4)=\frac16$,

(ii) calculate the values of p and q, and

(iii) find the probability distribution of S.