Review practice · Section 1 of 2

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 marks

Find E(X) and show that Var(X)=0.49.2 marks

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 marks

(ii) Find $E(X)$.2 marks

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 marks

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 marks

Two independent observations of X are taken. Find the probability that one is 2 and the other is at most 4.2 marks

An unbiased ordinary die is thrown. If its score is not 3, the result is that score. If it is 3, the die is thrown again and the result is the absolute difference of the two scores. Let R denote the result. Show that $P(R=0)=1/36$ and $P(R=1)=2/9$, and find the distribution of R.6 marks

Find

(i) the mean result,2 marks

(ii) the probability that the difference between $(R-1)^2$ and 3 is at most the mean from (i).4 marks