Task Sets: Calculate the expected value of a random variable - Common Core Math - HS Statistics and Probability

Warm-Up: Distributions and Weighted Means

These problems review probability distributions and the weighted-average idea you already know.

A random variable $X$ has the probability distribution below.

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

Which expression correctly sets up the expected value $E(X)$ ?

$0 \cdot \tfrac{1}{4} + 1 \cdot \tfrac{1}{2} + 2 \cdot \tfrac{1}{4}$

$\dfrac{0 + 1 + 2}{3}$

$\dfrac{1}{4} + \dfrac{1}{2} + \dfrac{1}{4}$

$0 + 1 + 2$

For a probability distribution, the mean of the distribution, written $\mu$ , is the same quantity as which of the following?

The expected value $E(X)$

The most likely value (the mode)

The number of possible values

The largest value the variable can take

A fair four-sided die shows the values $1, 2, 3, 4$ , each with probability $\tfrac{1}{4}$ . Because all outcomes are equally likely, $E(X)$ is just the plain average of the values. Compute $E(X)$ .

Fluency Practice

Compute each expected value as an explicit weighted sum. Write each term as value times its probability before adding.

A spinner has three regions with the distribution below.

$x$	5	8	20
$P(X=x)$	1/2	1/4	1/4

Which computation gives the correct expected value $E(X)$ ?

$5 \cdot \tfrac{1}{2} + 8 \cdot \tfrac{1}{4} + 20 \cdot \tfrac{1}{4} = 9.5$

$5 \cdot \tfrac{1}{3} + 8 \cdot \tfrac{1}{3} + 20 \cdot \tfrac{1}{3} = 11$

$\dfrac{5 + 8 + 20}{3} = 11$

$5 + 8 + 20 = 33$

A spinner is divided so that the value 1 has probability $\tfrac{3}{4}$ and the value 10 has probability $\tfrac{1}{4}$ .

$x$	1	10
$P(X=x)$	3/4	1/4

Compute the expected value $E(X)$ .

The number of pets $X$ owned by a randomly chosen student has the distribution below.

$x$	0	1	2	3
$P(X=x)$	0.4	0.3	0.2	0.1

Compute the mean of the distribution, $\mu = E(X)$ .

A carnival game pays out an amount $X$ per play according to the distribution below.

$x$ (dollars)	0	2	5
$P(X=x)$	0.6	0.3	0.1

If a player plays this game many, many times, what is the long-run average payout per play (in dollars)? Compute $E(X)$ .

In a game the net result $X$ is $+4$ dollars with probability $\tfrac{1}{4}$ and $-2$ dollars (a loss) with probability $\tfrac{3}{4}$ .

$x$ (dollars)	+4	-2
$P(X=x)$	1/4	3/4

Compute the expected net result $E(X)$ , in dollars. Keep the sign.

Exercises: Calculate the Expected Value of a Random Variable

Warm-Up: Distributions and Weighted Means

Fluency Practice