Practice section 5 of 7

A biased tetrahedral die is thrown onto a table; X is the number on the face in contact with the table. Its distribution is

x	$P(X=x)$
−1	$1/8$
0	$1/2$
2	$1/8$
3	$1/4$

(i) $X_1$ and $X_2$ are independent observations of X, and $Y=X_1+X_2$. Show that $P(Y=2)=3/16$.

(ii) A player pays 2 dollars to throw two such dice. The player receives 16 dollars if the maximum score is −1, or 3 dollars if the sum is prime, and receives nothing otherwise. Find the expected gain.

In a game, 3 red balls and 7 white balls are placed in a bag. A player draws 3 balls at random and without replacement. The player scores 2 points for every red ball that is drawn, and 1 point for every white ball that is drawn. The total score X is obtained by adding up the scores of the 3 balls.

Find the probability distribution of X. [3]

Find E(X) and show that Var(X)=0.49. [2]

An unbiased six-sided die is rolled twice. X is the higher value when they differ, or their common value when equal. Its distribution is $P(X=r)=k(2r-1)$ for r=1,2,3,4,5,6.

(i) Find the exact value of k as a fraction in simplest form. [2]

(ii) Find $E(X)$. [2]

One round consists of two rolls, with X as the score. A player plays three rounds.

(iii) Find the probability that the total score is 16. [2]

An unbiased disc has one dot on one side and two dots on the other. The disc and an unbiased ordinary die are thrown; X is the sum of the disc dots and the dots on top of the die. Tabulate the distribution of X.

Show that $P(X\ge4\mid X\le7)=8/11$. Write down $E(X)$ and show that $\operatorname{Var}(X)=19/6$.

Two independent observations $X_1,X_2$ are taken.

(i) Find $\operatorname{Var}(X_1-X_2)$.

(ii) Find $P(X_1-X_2\ge5)$.

X takes values −2,0,2 with probabilities 1/4,1/2,1/4 respectively. Find $\operatorname{Var}(X)$ and $E(|X|)$.

$Y=X_1+X_2$, where $X_1,X_2$ are independent observations of X. Find the distribution of Y, $\operatorname{Var}(Y)$ and $E(Y+3)$.

X has the following distribution.

x	$P(X=x)$
2	0.2
3	a
4	b
5	0.45

Given $E(|X-4|)=11/10$, find a and b. [3]

Two independent observations of X are taken. Find the probability that one is 2 and the other is at most 4. [2]