Expected Value — Practice Quiz

A Probability cheat sheet for Expected Value — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Expected Value (Discrete)

$$E(X) = \sum_{i} x_i \cdot P(x_i)$$

Linearity of Expectation

$$E(aX + b) = aE(X) + b$$

Where: \(a, b\) = constants

Variance

$$\text{Var}(X) = E(X^2) - [E(X)]^2$$

Standard Deviation

$$\sigma(X) = \sqrt{\text{Var}(X)}$$

Practice quiz

  1. A random variable $X$ has the following probability distribution: $P(X=1) = 0.2$, $P(X=2) = 0.5$, $P(X=3) = 0.3$. What is the expected value $E(X)$?

    • $2.1$
    • $2.0$
    • $2.2$
    • $1.9$

    Answer: $2.1$

  2. If $E(X) = 5$, what is $E(3X - 2)$?

    • $13$
    • $15$
    • $10$
    • $17$

    Answer: $13$

  3. Given $E(X) = 4$ and $E(X^2) = 20$, what is the variance $\text{Var}(X)$?

    • $4$
    • $16$
    • $20$
    • $0$

    Answer: $4$

  4. If the variance of a random variable $X$ is $\text{Var}(X) = 25$, what is its standard deviation $\sigma(X)$?

    • $5$
    • $25$
    • $12.5$
    • $0.2$

    Answer: $5$

  5. A discrete random variable $Y$ has the following probability distribution: $y_i$: $0$, $1$, $2$ $P(y_i)$: $0.3$, $0.4$, $0.3$ What is $E(Y)$?

    • $1.0$
    • $0.9$
    • $1.1$
    • $0.8$

    Answer: $1.0$

  6. If $E(2X + 5) = 15$, what is $E(X)$?

    • $5$
    • $10$
    • $2.5$
    • $7.5$

    Answer: $5$

  7. For a random variable $X$ with $P(X=0) = 0.5$ and $P(X=1) = 0.5$, what is $\text{Var}(X)$?

    • $0.25$
    • $0.5$
    • $0.75$
    • $1.0$

    Answer: $0.25$

  8. A random variable $Z$ has $P(Z= -1) = 0.4$ and $P(Z=1) = 0.6$. What is $\sigma(Z)$?

    • $\sqrt{0.96}$
    • $0.96$
    • $0.48$
    • $\sqrt{0.48}$

    Answer: $\sqrt{0.96}$

  9. Which of the following statements about variance is true?

    • $\text{Var}(X+c) = \text{Var}(X)$ for any constant $c$.
    • $\text{Var}(cX) = c \text{Var}(X)$ for any constant $c$.
    • $\text{Var}(X)$ can be negative.
    • $\text{Var}(X) = E(X^2) + [E(X)]^2$.

    Answer: $\text{Var}(X+c) = \text{Var}(X)$ for any constant $c$.

  10. A random variable $X$ has the following probability distribution: $P(X=0) = \frac{1}{3}$ and $P(X=2) = \frac{2}{3}$. What is the standard deviation $\sigma(X)$?

    • $\sqrt{\frac{8}{9}}$
    • $\frac{8}{9}$
    • $\sqrt{\frac{2}{3}}$
    • $\frac{2}{3}$

    Answer: $\sqrt{\frac{8}{9}}$

Select a subject

Select a subject from the left panel to begin exploring formulas.