Developing Empirical Probability Distributions, Expected Value - Statistics and Probability

1. What is the expected value formula for a discrete random variable $X$?

   $E(X)=\sum x\,P(x)$. Multiply each value by its probability and sum all products.

2. What is the sample space for $X=\text{number of TV sets per household}$ if observed values are $0$ to $5$?

   $\{0,1,2,3,4,5\}$. Sample space contains all possible values the random variable can take.

3. What is $E(X)$ for $X\in\{1,2,3\}$ with $P=\{0.2,0.5,0.3\}$?

   $E(X)=2.1$. $1(0.2)+2(0.5)+3(0.3)=0.2+1.0+0.9=2.1$

4. Find $E(X)$ if $P(X=1)=0.25$, $P(X=2)=0.50$, $P(X=3)=0.25$.

   $1(0.25)+2(0.50)+3(0.25)=2$. Weighted average: $1(0.25)+2(0.50)+3(0.25)=0.25+1+0.75$.

5. What is $E(X)$ for $X\in\{0,1,2,3\}$ with $P=\{0.1,0.2,0.3,0.4\}$?

   $E(X)=2.0$. $0(0.1)+1(0.2)+2(0.3)+3(0.4)=0+0.2+0.6+1.2=2.0$

6. What two conditions must a discrete probability distribution $P(X=x)$ satisfy?

   $0\le P(X=x)\le 1$ and $\sum P(X=x)=1$. Probabilities must be between 0 and 1, and sum to 1 for all outcomes.

7. Identify the random variable in: "number of TV sets per household" for a probability distribution.

   $X=\text{number of TV sets in a randomly selected household}$. Random variable represents the quantity being measured in each trial.

8. Find and correct the error: "$E(X)=\sum P(X=x)$" for a discrete random variable.

   Correct: $E(X)=\sum x\,P(X=x)$. Error: formula missing the $x$ values; must multiply $x$ by $P(X=x)$.

9. What is the expected value if $X$ takes values $0,1,2$ with probabilities $0.2,0.5,0.3$?

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

10. What is the expected value if a distribution is $P(0)=0.25$, $P(2)=0.75$?

    $E(X)=1.5$. $0(0.25)+2(0.75)=0+1.5=1.5$

11. What does $E(X)$ represent in context for TVs per household?

    Long-run average TVs per household. The average number of TVs we expect to find per household.

12. What is the expected number in $100$ households if $P(0)=0.2$, $P(1)=0.5$, $P(2)=0.3$?

    $110$. $E(X)=0(0.2)+1(0.5)+2(0.3)=1.1$, so $100(1.1)=110$

13. What is $E(X)$ if $X\in\{0,2\}$ with $P(X=0)=0.6$ and $P(X=2)=0.4$?

    $0(0.6)+2(0.4)=0.8$. Only two outcomes: $0$ contributes nothing, $2$ contributes $2(0.4)$.

14. What is the expected number of TV sets in $100$ households if $E(X)=1.7$ sets per household?

    $100\cdot 1.7=170$ sets. Multiply expected value per household by number of households.

15. Identify the random variable in: "number of TV sets in a randomly selected household."

    $X=\text{number of TV sets in one household}$. The variable counts TV sets in a single household.

16. What is $E(X)$ for $X\in\{0,1,2\}$ with empirical counts $10,30,60$ out of $100$?

    $0(0.1)+1(0.3)+2(0.6)=1.5$. Convert counts to probabilities: $\frac{10}{100}=0.1$, $\frac{30}{100}=0.3$, $\frac{60}{100}=0.6$.

17. What is an empirical probability for an outcome with frequency $f$ in $n$ trials?

    $P(\text{outcome})=\frac{f}{n}$. Divides frequency of occurrence by total number of trials.

18. Find the expected number in $200$ households if $E(X)=2.3$ TV sets per household.

    $200\cdot 2.3=460$ sets. Scale up single household expectation to 200 households.

19. Identify the missing probability if $P(X=0)=0.1$, $P(X=1)=0.4$, and $P(X=2)=?$ for a distribution.

    $P(X=2)=1-0.1-0.4=0.5$. Probabilities must sum to 1, so missing probability is $1-0.1-0.4$.

20. What is $E(X)$ if $X\in\{0,1,2,3\}$ with probabilities $0.1,0.2,0.3,0.4$?

    $0(0.1)+1(0.2)+2(0.3)+3(0.4)=2.0$. Sum products: $0+0.2+0.6+1.2=2.0$.

21. What is the expected count in $n$ trials if the expected value per trial is $E(X)$?

    $\text{Expected total}=n\,E(X)$. Multiply expected value per trial by number of trials.

22. State the formula for the expected total over $n$ independent observations of $X$ with mean $E(X)$.

    $E\left(\sum_{i=1}^{n} X_i\right)=nE(X)$. Expected sum of $n$ independent observations is $n$ times single expectation.

23. What must be true about the probabilities in a discrete probability distribution?

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

24. Identify the error: a "distribution" has $P(0)=0.4$, $P(1)=0.5$, $P(2)=0.3$.

    $\sum P(x)=1.2\ne 1$. Probabilities sum to more than 1, violating distribution rules.

25. What is the expected total in $100$ trials if $E(X)=1.2$ per trial?

    $120$. $100 \times 1.2 = 120$

26. What is the expected count in $n$ trials if the probability of success is $p$?

    $E(\text{count})=np$. Expected count in $n$ trials equals trials times success probability.

27. What is the expected number in $100$ households if $E(X)=2.3$ TVs per household?

    $230$. $100 \times 2.3 = 230$

28. Find the missing probability if $P(0)=0.12$, $P(1)=0.38$, $P(2)=0.27$, $P(3)=0.15$.

    $P(4)=0.08$. $1-0.12-0.38-0.27-0.15=0.08$

29. What is $E(X)$ if $P(X=0)=0.1$, $P(X=1)=0.6$, and $P(X=2)=0.3$?

    $E(X)=1.2$. $0(0.1)+1(0.6)+2(0.3)=0+0.6+0.6=1.2$

30. State the formula for the expected value of a discrete random variable $X$.

    $E(X)=\sum x\,P(X=x)$. Expected value is the weighted average using probabilities as weights.

31. What is $E(X)$ if $X$ is always $4$ (that is, $P(X=4)=1$)?

    $E(X)=4$. When a variable is constant, its expected value equals that constant.

32. What is the empirical probability if $18$ of $200$ households have $3$ TVs?

    $P(X=3)=0.09$. $\frac{18}{200}=0.09$

33. What is the empirical probability if $7$ of $50$ households have $0$ TVs?

    $P(X=0)=0.14$. $\frac{7}{50}=0.14$

34. What is the empirical distribution if counts for $X=0,1,2$ are $5,15,30$ out of $50$?

    $P(0)=0.1,\;P(1)=0.3,\;P(2)=0.6$. Divide each count by total 50: $\frac{5}{50}=0.1$, $\frac{15}{50}=0.3$, $\frac{30}{50}=0.6$.

35. What is the sample space for $X$ = number of TV sets per household (discrete, nonnegative)?

    $\{0,1,2,3,\dots\}$. All possible non-negative integer counts.

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