Statistics Flashcards: Developing Theoretical Probability Distributions Expected Value

What this deck covers

This deck focuses on Developing Theoretical Probability Distributions Expected Value, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: State the expected value formula for $X\sim\text{Binomial}(n,p)$.

Answer: $E(X)=np$. Mean of binomial distribution is number of trials times success probability.

Flashcard 2: What is the expected value of a constant random variable $X=c$?

Answer: $E(X)=c$. A constant has no variability, so its expected value equals itself.

Flashcard 3: What is $P(X=1)$ for $X\sim\text{Binomial}(5,\frac{1}{4})$?

Answer: $\binom{5}{1}\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^4$. Uses binomial formula with $n=5$, $k=1$, $p=\frac{1}{4}$.

Flashcard 4: What is the theoretical probability of a correct guess on a $4$-choice multiple-choice question?

Answer: $\frac{1}{4}$. One correct choice out of four equally likely options.

Flashcard 5: State the expected value for $X\sim\text{Binomial}(n,p)$.

Answer: $E(X)=np$. Average number of successes equals trials times success probability.

Flashcard 6: What is $P(X=0)$ when guessing on $5$ questions with $p=\frac{1}{4}$?

Answer: $\left(\frac{3}{4}\right)^5$. All 5 wrong: $(1-p)^5$ where $p=\frac{1}{4}$.

Flashcard 7: What is $P(X=5)$ when guessing on $5$ questions with $p=\frac{1}{4}$?

Answer: $\left(\frac{1}{4}\right)^5$. All 5 correct: $p^5$ where $p=\frac{1}{4}$.

Flashcard 8: State the linearity rule for expected value: what is $E(aX+b)$?

Answer: $E(aX+b)=aE(X)+b$. Constants factor out; adding $b$ shifts expected value by $b$.

Flashcard 9: Which expression gives $P(X\ge 3)$ for $X\sim\text{Binomial}(5,\frac{1}{4})$?

Answer: $1-[P(X=0)+P(X=1)+P(X=2)]$. Complement rule: $P(X\ge 3)=1-P(X\le 2)$.

Flashcard 10: What is the expected score if each correct answer is worth $2$ points and $X$ is # correct?

Answer: $E(2X)=2E(X)$. Linearity of expectation: constant multiplier scales the expected value.

Flashcard 11: State the binomial probability formula for $P(X=k)$ when $X\sim\text{Binomial}(n,p)$.

Answer: $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$. Counts ways to get $k$ successes times probability of each arrangement.

Flashcard 12: Find $E(X)$ if $X\in\{0,1,2\}$ with $P(0)=0.2$, $P(1)=0.5$, $P(2)=0.3$.

Answer: $E(X)=0(0.2)+1(0.5)+2(0.3)=1.1$. Multiply each value by its probability and sum: $0.5+0.6=1.1$.

Flashcard 13: What two conditions must a probability distribution satisfy for values $x_i$ with probabilities $p_i$?

Answer: $p_i \ge 0$ and $\sum p_i = 1$. Probabilities must be non-negative and sum to 1 for a valid distribution.

Flashcard 14: If score is $S=2X$, what is $E(S)$ in terms of $E(X)$?

Answer: $E(S)=2E(X)$. Expected value scales linearly with constant multiplier.

Flashcard 15: If each correct is $+4$ points and wrong is $0$, what is expected score for $5$ questions with $p=\frac{1}{4}$?

Answer: $E(S)=4\cdot\frac{5}{4}=5$. Score is $4X$, so $E(S)=4E(X)=4\cdot\frac{5}{4}$.

Flashcard 16: What is $P(X=5)$ for guessing on $5$ questions with $p=\frac{1}{4}$?

Answer: $\left(\frac{1}{4}\right)^5$. All 5 correct means $p^5$ where $p=\frac{1}{4}$.

Flashcard 17: What is the expected number correct when guessing on $5$ questions with $p=\frac{1}{4}$?

Answer: $E(X)=5\cdot\frac{1}{4}=\frac{5}{4}$. For binomial, expected value equals $np$.

Flashcard 18: What is $P(X=4)$ for $X\sim\text{Binomial}(5,\frac{1}{4})$?

Answer: $\binom{5}{4}\left(\frac{1}{4}\right)^4\left(\frac{3}{4}\right)$. Uses binomial formula with $n=5$, $k=4$, $p=\frac{1}{4}$.

Flashcard 19: What two conditions must a probability distribution for discrete $X$ satisfy?

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

Flashcard 20: What is $P(X=0)$ for guessing on $5$ questions with $p=\frac{1}{4}$?

Answer: $\left(\frac{3}{4}\right)^5$. All 5 wrong means $(1-p)^5$ where $p=\frac{1}{4}$.

Flashcard 21: What is the expected score for scheme $S=1X+0(5-X)$ when guessing on $5$ with $p=\frac{1}{4}$?

Answer: $E(S)=E(X)=\frac{5}{4}$. Only correct answers score, so $S=X$ and $E(S)=E(X)$.

Flashcard 22: What is the expected score for $5$ questions, $2$ points each correct, guessing with $p=\frac{1}{4}$?

Answer: $2\cdot\frac{5}{4}=\frac{5}{2}$. Apply $E(2X)=2E(X)$ with $E(X)=\frac{5}{4}$.

Flashcard 23: Identify the distribution of $X$ = number correct when guessing on $n$ independent questions with success prob $p$.

Answer: $X\sim\text{Binomial}(n,p)$. Counting successes in $n$ independent trials follows binomial distribution.

Flashcard 24: If each correct is $+1$ and each wrong is $-\frac{1}{3}$, what is expected score for $5$ questions with $p=\frac{1}{4}$?

Answer: $5\left(\frac{1}{4}-\frac{1}{3}\cdot\frac{3}{4}\right)=0$. Expected gain per question is $1\cdot p-\frac{1}{3}(1-p)=0$ when $p=\frac{1}{4}$.

Flashcard 25: State the linearity rule for expected value of $Y=aX+b$.

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

Flashcard 26: If each correct is $+1$ point and no penalty, what is expected score for $5$ questions with $p=\frac{1}{4}$?

Answer: $E(S)=E(X)=\frac{5}{4}$. Score equals number correct, so $E(S)=E(X)=np$.

Flashcard 27: What is $P(X=2)$ for $X\sim\text{Binomial}(5,\frac{1}{4})$?

Answer: $\binom{5}{2}\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^3$. Uses binomial formula with $n=5$, $k=2$, $p=\frac{1}{4}$.

Flashcard 28: What is $P(X=2)$ for guessing on $5$ questions with $p=\frac{1}{4}$?

Answer: $\binom{5}{2}\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^3$. Choose 2 positions for successes, multiply by $p^2$ and $(1-p)^3$.

Flashcard 29: What is $P(X=3)$ for $X\sim\text{Binomial}(5,\frac{1}{4})$?

Answer: $\binom{5}{3}\left(\frac{1}{4}\right)^3\left(\frac{3}{4}\right)^2$. Uses binomial formula with $n=5$, $k=3$, $p=\frac{1}{4}$.

Flashcard 30: Identify the distribution of $X$ = number correct when guessing on $n$ 4-choice questions.

Answer: $X\sim\text{Binomial}(n,\frac{1}{4})$. Counting successes in $n$ independent trials with success probability $\frac{1}{4}$.

Flashcard 31: What is $P(X=1)$ for guessing on $5$ questions with $p=\frac{1}{4}$?

Answer: $\binom{5}{1}\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^4$. Choose 1 position for success, multiply by $p$ and $(1-p)^4$.

Flashcard 32: State the formula for the expected value $E(X)$ of a discrete random variable.

Answer: $E(X)=\sum x_i p_i$. Sum each outcome times its probability to find the average value.

Flashcard 33: What is the expected number correct when guessing on $5$ questions with $p=\frac{1}{4}$?

Answer: $E(X)=5\cdot\frac{1}{4}=\frac{5}{4}$. For binomial, $E(X)=np=5\cdot\frac{1}{4}$.

Flashcard 34: What is the expected percent score if percent $=20X$ for $5$ questions and $p=\frac{1}{4}$?

Answer: $E(20X)=20\cdot\frac{5}{4}=25$. Each correct is worth 20%, so multiply expected correct by 20.

Flashcard 35: State the binomial probability formula for $P(X=k)$ when $X\sim\text{Binomial}(n,p)$.

Answer: $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$. Gives probability of exactly $k$ successes in $n$ trials.

Flashcard 36: State the formula for the expected value of a discrete random variable $X$.

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

Flashcard 37: What is $P(\text{correct})$ when guessing on one 4-choice multiple-choice question?

Answer: $\frac{1}{4}$. One correct choice out of four equally likely options.