JC2 H2 Mathematics: Discrete Random Variables

At most three boxes are opened

One of six boxes contains a prize. Boxes are opened at random without replacement until the prize is found, with at most three boxes opened. Let X be the number of empty boxes opened. Find the distribution of X.

The possible values are 0, 1, 2 and 3. The value 2 means the prize is found third; the value 3 means all three attempts fail.

The first three probabilities are equal because the prize’s position in a random ordering is equally likely to be any of the six positions. Positions 4–6 all lead to X = 3.

The standard deviation is $\sqrt{4/3}\approx1.155$. Therefore $P(X\leq\sigma)=P(X\leq1)=1/3$. X takes integer values: do not add a fraction of the probability at 2.

Recover three missing probabilities

X takes values $-2,-1,0,1,3$ with probabilities $0.1,a,0.3,b,c$. Given $E(X)=0.9$ and $\operatorname{Var}(X)=2.99$, find a, b and c.

Three unknowns need three equations. The variance gives the second moment:

Subtract the first equation from the simplified third:

All probabilities are valid and sum to one. In $E(X^2)$, square the values of X; do not square their probabilities.

A squared expression is not a variance

Given $E(X)=4$ and $E[(X-3)^2]=40$, find $\operatorname{Var}(X)$.

A variance uses the mean as the centre. Here X − 3 has mean 1, so subtract its squared mean:

Subtracting a constant does not change spread. Equivalently expand the square, find $E(X^2)=55$, then subtract $4^2$.

The smaller of two die scores

Two independent fair four-sided dice show 1, 2, 3 or 4. Let M be the smaller score, using the common score if they are equal. Find its distribution.

The event $M\geq m$ requires both scores to be at least m. Subtract the next tail to isolate one value:

The 16 ordered die outcomes are equally likely; the four possible values of the minimum are not equally likely.

An absolute difference uses both directions

Let $P(X=x)=kx$, for $x=1,2,3,4$. Find k, the mean and variance, then $P(|X_1-X_2|\geq2)$ for two independent observations.

Pairs with a positive difference of at least 2 are $(3,1),(4,1),(4,2)$. The reversed pairs have the same probability by identical distributions and independence:

Write q for the required probability.

A signed difference $X_1-X_2\geq2$ would give 0.15. The wording “absolute difference” decides whether reversed pairs belong.

Condition on the sum of several scores

Three independent games each give score 0, 1 or 2 with probabilities 0.2, 0.5 and 0.3. Given a total score of 3, find the probability that the first score was 2.

The denominator contains $(1,1,1)$ and the six permutations of $(0,1,2)$. For a first score of 2, only $(2,0,1)$ and $(2,1,0)$ remain.

Write $a=P(T=3)$, $b=P(X_1=2,T=3)$, and q for the required conditional probability.

The total condition destroys the original independence between scores. Use their independence to calculate unconditional path probabilities before taking the conditional ratio.

Translate the score into money before judging the game

Use the distribution $P(X=x)=x/10$ from the preceding example. A game pays a gross prize of $3X and costs $8 to enter. Let Y be the player’s net gain. Find the mean and variance of Y, then approximate the probability that the mean net gain in 50 independent games exceeds $1.50.

The expected net gain is $1 per game, so the game favours the player in the long run. A positive expectation does not guarantee a profit in one session. For 50 independent games, by the Central Limit Theorem,

The normal model applies to the mean over many games. It does not make the four-valued score X normally distributed. More examples of totals, means and sample size appear in the Sampling guide.