Statistics Quiz: Calculating Expected Value Of Random Variables

A raffle ticket has the following net winnings (in dollars). Let $X$ be the net winnings from buying one ticket. $P(X=-2)=\tfrac{3}{4}$ and $P(X=6)=\tfrac{1}{4}$. What is the expected value of $X$ (in dollars)? (Note: the expected value may not be a possible outcome.)

A prize pays $10 with probability 0.25, otherwise $0. What ticket price makes expected net gain zero?

A fair coin is flipped 4 times. For each head you win $3; for each tail you lose $2. What is the expected net gain?

A game costs $3 to play. It pays $8 with probability 0.4, otherwise $0. What is the expected net gain?

A random variable X has mean 7. What is the expected total over 100 independent observations of X?

Let X be a profit in dollars taking values -$2, $1, and $5 with probabilities 0.4, 0.3, and 0.3. Find E(X).

A simple game defines a random variable $X$ as the number of points scored in one round. The distribution is: $P(X=1)=0.3$, $P(X=2)=0.4$, $P(X=5)=0.3$. What is the expected value of $X$ (in points)? (Compute $\sum x\,P(x)$; the expected value may not be a possible outcome.)

A box contains 3 red balls and 1 blue ball. One ball is drawn at random. Let $X$ be the value of the draw, where $X=2$ if the ball is red and $X=8$ if the ball is blue. The distribution is: $P(X=2)=\tfrac{3}{4}$, $P(X=8)=\tfrac{1}{4}$. What is the expected value of $X$? (Compute $\sum x\,P(x)$; the expected value may not be a possible outcome.)

A spinner game defines a random variable $X$ as the prize amount (in dollars) from one spin. The distribution is: $P(X=0)=\tfrac{1}{2}$, $P(X=4)=\tfrac{1}{4}$, and $P(X=10)=\tfrac{1}{4}$. What is the expected value of $X$ (in dollars)? (Compute $\sum x\,P(x)$; the expected value may not be a possible outcome.)

A fair coin is flipped twice. Define the random variable $X$ as the number of heads obtained. The distribution is: $P(X=0)=\tfrac{1}{4}$, $P(X=1)=\tfrac{1}{2}$, $P(X=2)=\tfrac{1}{4}$. What is the expected value of $X$? (Compute $\sum x\,P(x)$; the expected value may not be a possible outcome.)

A die-based game works like this: roll a fair six-sided die once. Let $X$ be the payout in dollars, where $X=0$ if the roll is 1–3, $X=3$ if the roll is 4–5, and $X=9$ if the roll is 6. What is the expected value of $X$ (in dollars)? (Note: the expected value may not be a possible payout.)

A coupon game gives a discount in dollars. Let $X$ be the discount from one play. The distribution is: $P(X=1)=\tfrac{1}{4}$, $P(X=3)=\tfrac{1}{2}$, $P(X=5)=\tfrac{1}{4}$. What is the expected value of $X$ (in dollars)? (Note: the expected value may not be a possible outcome.)

A survey records how many days per week a student exercises. Let $X$ be the number of exercise days per week. The distribution is: $P(X=0)=\tfrac{1}{4}$, $P(X=2)=\tfrac{1}{2}$, $P(X=4)=\tfrac{1}{4}$. What is the expected value of $X$ (in days)? (Note: the expected value may not be a possible outcome.)

A simple game pays out dollars based on a card draw. Let $X$ be the payout from one draw. The distribution is: $P(X=0)=\tfrac{2}{5}$, $P(X=5)=\tfrac{2}{5}$, $P(X=10)=\tfrac{1}{5}$. What is the expected value of $X$ (in dollars)? (Note: the expected value may not be a possible payout.)

A game awards $X$ points with probabilities: $P(X=0)=\tfrac{1}{4}$, $P(X=4)=\tfrac{1}{2}$, and $P(X=8)=\tfrac{1}{4}$. What is the expected value of $X$? Give your answer in points.

A raffle ticket yields winnings $X$ (in dollars) with probabilities: $P(X=-1)=\tfrac{3}{4}$ and $P(X=5)=\tfrac{1}{4}$. (A negative value means you lose money overall.) What is the expected value of $X$? Give your answer in dollars.

A bag contains cards labeled 1, 2, and 5. One card is drawn at random. Let $X$ be the number on the card. The probabilities are $P(X=1)=\tfrac{1}{2}$, $P(X=2)=\tfrac{1}{3}$, and $P(X=5)=\tfrac{1}{6}$ (they sum to 1). What is the expected value of $X$?

A survey records the number of minutes a student spends reading on a given day. Let $X$ be the number of minutes. The distribution is:

$P(X=10)=\tfrac{1}{4}$, $P(X=20)=\tfrac{1}{2}$, $P(X=30)=\tfrac{1}{4}$.

What is the expected value of $X$? Give your answer in minutes.

A raffle ticket has the following net winnings (in dollars). Let $X$ be the net winnings from one ticket: $P(X=-1)=\tfrac{3}{4}$, $P(X=3)=\tfrac{1}{4}$. What is the expected value of $X$ in dollars? (It may not be an outcome.)

A single fair six-sided die is rolled once. Let $X$ be the number shown. The distribution is $P(X=k)=\tfrac{1}{6}$ for $k=1,2,3,4,5,6$. What is the expected value of $X$? (Compute $\sum x\,P(x)$; the expected value may not be a possible outcome.)