Topic 4.7: Discrete Random Variables Questions

A discrete random variable $X$ has $P(X=x)=\frac{x+3}{15}$ for $x=1,2,3$. Show that this is a valid probability distribution.

The random variable X represents a player's net gain in dollars in a game where a game costs $4 to play and the prize for the third outcome is $20. The probability distribution of X is

x	25	B − 4	A	−4
P(X = x)	0.0512	0.0512	0.1728	p

where the third outcome earns the $20 prize.

Write down the value of A.

A spinner is spun until it stops on red. Let Y be the number of spins needed, 1 ≤ Y ≤ 6. It is given that P(Y = 1) = k and that each further spin multiplies the probability by 2/3, so P(Y = n + 1) = (2/3) P(Y = n) for 1 ≤ n ≤ 5, where k ∈ ℝ⁺. Let a = P(Y = 2) and b = P(Y = 5).

Write down, in terms of k,

(i) a;

(ii) b.

The discrete random variable X takes the values 1, 2, 3, 4, 5 and 6, with P(X = 1) = k and P(X = r + 1) = (1/3) P(X = r) for 1 ≤ r ≤ 5, where k ∈ ℝ⁺. In the probability distribution table, P(X = 2) = a and P(X = 4) = b.

Write down an expression in terms of k for

(i) a;

(ii) b.

A raffle ticket costs $5. A ticket can win $30, B dollars or $50, or nothing. Let X be the net gain, in dollars, from buying one ticket; its distribution is

x	25	B − 5	A	−5
P(X = x)	0.02	0.03	0.05	0.9

where A is the net gain from the $50 prize.

Write down the value of A.

A spinner pays out X dollars with $P(X=0)=0.5,P(X=2)=0.3,P(X=5)=0.2$. Find the expected payout.

The number of goals X a team scores in a match has the probability distribution P(X = 0) = 0.35, P(X = 1) = 0.4, P(X = 2) = 0.2, P(X = 3) = 0.05.

Interpret, in context, the most likely outcome of the random variable.

A box holds three bulbs. The number of faulty bulbs Y in a box has the distribution P(Y = 0) = 0.72, P(Y = 1) = 0.24, P(Y = 2) = 0.035, P(Y = 3) = 0.005.

Interpret, in context, the most likely outcome of the random variable.

Write down the value of $k$.

A discrete random variable $X$ takes the values $1$, $2$ and $3$. $P(X=1)=0.6$, $P(X=2)=m$ and $P(X=3)=n$. State the range of possible values of $m$.

A discrete random variable $X$ has $P(X=1)=k-0.15$, $P(X=2)=0.3$, $P(X=3)=0.5-2k^2$ and $P(X=4)=0.27$. Show that $2k^2-k+0.08=0$.

A spinner pays out X dollars with P(X = 0) = 0.5, P(X = 2) = 0.3, P(X = 5) = 0.2. Find the expected payout.

In one round of a game a player scores 2, 4 or 6 points, with P(2) = 0.5, P(4) = 0.3, P(6) = 0.2.

Interpret, in context, the most likely outcome of the random variable.

X has $P(X=1)=0.2,P(X=2)=0.5,P(X=3)=0.3$. Find $E(X)$.

The number of days, $X$, that a library book is kept has the distribution $P(X=1)=0.15$, $P(X=2)=0.2$, $P(X=3)=0.35$, $P(X=4)=0.2$ and $P(X=5)=0.1$. Write down the mode of $X$.

Show that $E(X)=1$.

X has P(X = 1) = 0.2, P(X = 2) = 0.5, P(X = 3) = 0.3. Find E(X).

The random variable $Y$ takes the values $0$, $1$ and $2$. $P(Y=2)=\frac{1}{4}$ and $E(Y)=1$. The probability generating function of $Y$ is $G_Y(t)=P(Y=0)+P(Y=1)t+P(Y=2)t^2$. Find an expression for $G_Y(t)$.

A discrete random variable X has $P(X=x)=c(5-x)$ for $x=1,2,3,4$. Find c.

In a game you pay $3 to roll a fair die. You receive $x equal to the number rolled. Find the expected gain per game, and state whether it is worth playing.