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Statistics

Unit 7

Random Variables

6 topics · 24 practice questions · Free responseNot yet reviewed

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Notes

Discrete random variables

  • A discrete random variable has a list of values with probabilities that add to 1.
  • Expected value: $\displaystyle E(X) = \sum x\,P(x)$, the long-run average.
  • $E(aX + b) = aE(X) + b$ and $E(X + Y) = E(X) + E(Y)$. For independent variables, variances add.

Binomial distribution

  • Binomial setting (BINS): binary outcomes, independent trials, fixed number $n$, same $p$.
  • $P(X = k) = \binom{n}{k}p^k(1 - p)^{n-k}$. Mean $np$ and standard deviation $\displaystyle \sqrt{np(1 - p)}$.
  • For "at least one," use the complement.

Simulation

  • Assign random digits to outcomes in proportion to their probabilities, repeat many trials, and estimate with $\displaystyle \frac{\text{successes}}{\text{trials}}$.

Topics in this unit

Each topic has its own short notes and practice.

Practice

$X$ takes the values 10, 20, 30 and 40 with probabilities $0.25$, $0.35$, $p$ and $0.1$. What is $p$?
$X = 1, 2, 3, 4$ with probabilities $0.1, 0.2, 0.3, 0.4$. What is $E(X)$?
$E(X) = 4$ and $Y = 2X - 1$. What is $E(Y)$?
A binomial variable has $n = 50$ and $p = 0.4$. Its mean is:
Which variable is NOT binomial?
A binomial variable has $n = 2$ and $p = 0.2$. What is $P(X = 1)$?

Free response

Write your answer first, then open the scoring guide.

Free-response question

At a school fair, you roll a fair die. You win $12 if you roll a 6, win $3 if you roll a 4 or 5, and win $0 otherwise. Let $X$ be your winnings. It costs $4 to play.

(a) Write the probability distribution of $X$.

Show the answer and scoring guide

$\displaystyle P(X = 12) = \frac{1}{6}$, $\displaystyle P(X = 3) = \frac{2}{6} = \frac{1}{3}$, $\displaystyle P(X = 0) = \frac{3}{6} = \frac{1}{2}$. The probabilities add to 1.

Check off the points you earned:

(b) Find the expected winnings $E(X)$.

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$\displaystyle E(X) = 12\left(\frac{1}{6}\right) + 3\left(\frac{1}{3}\right) + 0\left(\frac{1}{2}\right) = 2 + 1 + 0 = 3$ dollars.

Check off the points you earned:

(c) Find the expected net gain per play after paying $4, and interpret it.

Show the answer and scoring guide

Expected net $= 3 - 4 = -1$ dollar. In the long run, a player loses about $1 per play on average, so the game is not fair.

Check off the points you earned:

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