Calculating Expected Value of Random Variables - Statistics and Probability

1. Find and correct the formula error: $E(X)=\sum P(X=x)$.

   Correct: $E(X)=\sum x\,P(X=x)$. Missing the multiplication by $x$ in each term.

2. Identify the condition that must hold for a valid discrete probability distribution.

   $\sum P(X=x)=1$ and each $P(X=x)\ge 0$. Probabilities must be non-negative and sum to 1.

3. Calculate $E(X)$ if $X=10$ with probability $0.1$ and $X=0$ otherwise.

   $E(X)=1$. $E(X) = 10(0.1) + 0(0.9) = 1$.

4. Compute $E(X+Y)$ given $E(X)=1.2$ and $E(Y)=-0.7$.

   $0.5$. $E(X+Y) = 1.2 + (-0.7) = 1.2 - 0.7$.

5. What does the expected value $E(X)$ represent in a probability distribution?

   The mean (long-run average) of the distribution. It's the average value if repeated infinitely.

6. Which expression gives $E(X)$ when outcomes are $x_1,\dots,x_n$ with probabilities $p_1,\dots,p_n$?

   $E(X)=\sum_{i=1}^{n} x_i p_i$. Alternative notation using indexed outcomes and probabilities.

7. Identify the expected value $E(X)$ if $P(X=2)=1$ (a certain outcome).

   $E(X)=2$. When only one outcome is possible, $E(X)$ equals that outcome.

8. State the additivity rule for expected value of a sum $X+Y$ (no independence required).

   $E(X+Y)=E(X)+E(Y)$. Expected values add even without independence.

9. Compute $E(X)$ for a fair coin where $X=1$ for heads and $X=0$ for tails.

   $E(X)=0.5$. Fair coin: $E(X) = 0(0.5) + 1(0.5)$.

10. Calculate $E(X)$ if $P(X=0)=0.2$ and $P(X=5)=0.8$.

    $E(X)=4$. $E(X) = 0(0.2) + 5(0.8) = 0 + 4 = 4$.

11. Compute the expected value of a fair six-sided die roll $X\in\{1,2,3,4,5,6\}$.

    $E(X)=3.5$. $E(X) = \frac{1+2+3+4+5+6}{6} = \frac{21}{6}$.

12. Identify the expected value for a fair game with outcomes as net gains and losses.

    $E(\text{net gain})=0$. Fair games have zero expected net gain/loss.

13. State the linearity rule for the expected value of $aX+b$.

    $E(aX+b)=aE(X)+b$. Linear transformations preserve linearity in expectation.

14. Calculate $E(X)$ if $X\in\{-1,1\}$ with $P(X=-1)=0.7$ and $P(X=1)=0.3$.

    $E(X)=-0.4$. $E(X) = -1(0.7) + 1(0.3) = -0.7 + 0.3 = -0.4$.

15. Compute $E(X)$ for $X\in\{-1,2\}$ with $P(X=-1)=0.6$ and $P(X=2)=0.4$.

    $E(X)=0.2$. $E(X) = -1(0.6) + 2(0.4) = -0.6 + 0.8$.

16. Compute $E(X)$ for $X\in\{1,3\}$ with $P(X=1)=\frac{1}{4}$ and $P(X=3)=\frac{3}{4}$.

    $E(X)=\frac{5}{2}$. $E(X) = 1(\frac{1}{4}) + 3(\frac{3}{4}) = \frac{1}{4} + \frac{9}{4} = \frac{10}{4}$.

17. Compute $E(X)$ for a uniform distribution on $\{2,4,6,8\}$.

    $E(X)=5$. $E(X) = \frac{2+4+6+8}{4} = \frac{20}{4} = 5$.

18. Calculate $E(X)$ for $X\in\{1,2,3\}$ with probabilities $\{0.2,0.5,0.3\}$.

    $E(X)=2.1$. $E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 0.2 + 1 + 0.9 = 2.1$.

19. What does it mean if a game has expected value $E(G)=0$ for the player's net gain?

    It is fair in the long run. Zero expected value means no advantage to either side.

20. State the shortcut for $E(X)$ when $X$ takes values $a$ and $b$ with probabilities $p$ and $1-p$.

    $E(X)=ap+b(1-p)$. Direct formula for two-outcome distributions.

21. Calculate $E(X)$ for a fair coin where $X=1$ for heads and $X=0$ for tails.

    $E(X)=0.5$. $E(X) = 0(0.5) + 1(0.5) = 0.5$.

22. Interpret $E(X)=2.4$ in a repeated-trials context.

    Average outcome approaches $2.4$ per trial in the long run. Over many trials, the average result converges to $E(X)$.

23. What is $E(c)$ if $c$ is a constant random variable (always equals $c$)?

    $E(c)=c$. A constant's expected value is itself.

24. What condition must a discrete probability distribution satisfy for $E(X)$ to be computed?

    $\sum P(X=x)=1$ and $P(X=x)\ge 0$. Probabilities must be non-negative and sum to 1.

25. What is the interpretation of $E(X)$ in a probability distribution for a discrete random variable?

    The long-run mean (average) outcome. Expected value represents the average if repeated infinitely.

26. Compute $E(X)$ for $X\in\{0,1,2\}$ with $P(0)=0.25$, $P(1)=0.5$, $P(2)=0.25$.

    $E(X)=1$. $E(X) = 0(0.25) + 1(0.5) + 2(0.25) = 0 + 0.5 + 0.5 = 1$.

27. Identify the expected value of net gain $G$ for a $2 ticket that pays $8 with probability $0.1$ and $0 otherwise.

    $E(G)=-1.2$ dollars. $E(G) = (8-2)(0.1) + (0-2)(0.9) = 0.6 - 1.8$.

28. Find $E(X)$ if $X$ takes values $0$ and $1$ with $P(X=1)=p$ and $P(X=0)=1-p$.

    $E(X)=p$. For Bernoulli: $E(X) = 0(1-p) + 1(p) = p$.

29. Compute expected winnings $E(W)$ if you win $10 with probability $0.3$ and lose $4 with probability $0.7$.

    $E(W)=0.2$ dollars. $E(W) = 10(0.3) + (-4)(0.7) = 3 - 2.8$.

30. Identify the expected value of a constant random variable $X=c$.

    $E(X)=c$. Constant values have expected value equal to the constant.

31. Compute $E(X)$ for $X\in\{0,1,2\}$ with probabilities $0.2,0.5,0.3$ respectively.

    $E(X)=1.1$. $E(X) = 0(0.2) + 1(0.5) + 2(0.3) = 0 + 0.5 + 0.6$.

32. If $E(X)=4$, what is $E(3X-2)$?

    $E(3X-2)=10$. $E(3X-2) = 3(4) - 2 = 12 - 2 = 10$.

33. State the expected value rule for a linear transformation $Y=aX+b$.

    $E(Y)=aE(X)+b$. Linear transformations scale and shift the expected value.

34. Which value must you compute to interpret the mean of a discrete probability distribution?

    The expected value $E(X)$. The mean of a distribution is its expected value.

35. State the formula for the expected value $E(X)$ of a discrete random variable.

    $E(X)=\sum x\,P(X=x)$. Sum each outcome times its probability.

36. Find $E(X)$ for a fair six-sided die roll with outcomes $1$ to $6$.

    $E(X)=3.5$. $E(X) = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5$.

37. State the formula for the expected value $E(X)$ of a discrete random variable $X$.

    $E(X)=\sum x\,P(X=x)$. Sum each outcome times its probability.

38. Compute $E(3X-5)$ given that $E(X)=4$.

    $7$. $E(3X-5) = 3E(X) - 5 = 3(4) - 5 = 12 - 5$.

39. Which value equals the mean of a discrete probability distribution: median, mode, or expected value?

    Expected value. Expected value and mean are the same for probability distributions.

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