Expected Value — Hard Practice Quiz

A Probability cheat sheet for Expected Value — every key formula with its symbols defined — plus a hard-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 discrete random variable $X$ has the following probability distribution: $P(X=1) = 0.2$, $P(X=2) = 0.3$, $P(X=3) = 0.5$. If a new random variable $Y$ is defined as $Y = 4X - 7$, what is the expected value of $Y$?

    • $2.2$
    • $9.2$
    • $1.2$
    • $0.2$

    Answer: $2.2$

  2. Given that $E(X) = 5$ and $E(X^2) = 30$, what is the standard deviation of the random variable $Y = -2X + 10$?

    • $2\sqrt{5}$
    • $\sqrt{5}$
    • $10$
    • $5\sqrt{2}$

    Answer: $2\sqrt{5}$

  3. A random variable $X$ has a variance of $\text{Var}(X) = 9$. If a new random variable $Z$ is defined as $Z = \frac{X}{3} - 5$, what is the variance of $Z$?

    • $1$
    • $3$
    • $9$
    • $0$

    Answer: $1$

  4. For a discrete random variable $X$, $P(X=0) = 0.4$, $P(X=1) = 0.3$, and $P(X=2) = 0.3$. Calculate $E(X^2)$ and then use it to find $\text{Var}(X)$.

    • $0.69$
    • $1.5$
    • $0.81$
    • $0.9$

    Answer: $0.69$

  5. If $E(X) = \mu$ and $\text{Var}(X) = \sigma^2$, what is $E((X - \mu)^2)$?

    • $\sigma^2$
    • $\mu^2$
    • $0$
    • $E(X^2)$

    Answer: $\sigma^2$

  6. A random variable $X$ has $E(X) = 4$ and $\sigma(X) = 3$. What is $E(X^2)$?

    • $25$
    • $16$
    • $9$
    • $7$

    Answer: $25$

  7. Consider a random variable $X$ with $E(X) = 0$. Which of the following statements is always true?

    • $E(X^2) = \text{Var}(X)$
    • $\text{Var}(X) = 0$
    • $X$ must be $0$
    • $E(X^2) < 0$

    Answer: $E(X^2) = \text{Var}(X)$

  8. A fair six-sided die is rolled. Let $X$ be the number rolled. If you win $2X-1$ dollars for each roll, what is the standard deviation of your winnings?

    • $\sqrt{\frac{35}{3}}$
    • $\sqrt{\frac{35}{12}}$
    • $2\sqrt{\frac{35}{12}}$
    • $3.5$

    Answer: $\sqrt{\frac{35}{3}}$

  9. If $E(X) = 10$ and $\text{Var}(X) = 4$, what is the expected value of $(X+1)^2$?

    • $125$
    • $104$
    • $121$
    • $100$

    Answer: $125$

  10. Let $X$ be a random variable. If $Y = aX + b$, where $a$ and $b$ are constants, derive an expression for $\text{Var}(Y)$ in terms of $\text{Var}(X)$.

    • $a^2\text{Var}(X)$
    • $a\text{Var}(X)+b$
    • $\text{Var}(X)+b^2$
    • $a^2\text{Var}(X)+b^2$

    Answer: $a^2\text{Var}(X)$

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