# Random Variables

STAT 20: Introduction to Probability and Statistics

## Agenda

• PS 8: time to work on it and then review
• Brief lecture on random variables
• Break
• Concept Questions
• PS 9

25:00

# Lecture

## Random variables

• Let $X$ be the number of heads in three tosses of a fair coin. What is the distribution of $X$? That is, what is $f(x) = P(X = x)$? What values will $x$ take?

• What about if $X$ is the number of heads in 3 tosses of a biased coin, where the chance of heads is $\frac{2}{3}$? Now what is $f(x) = P(X = x)$?

• Now suppose we toss a fair coin until the first time it lands heads, and let $X$ be the number of tosses (including the last one, which is the first time the coin lands heads). What is the pmf of $X$? Is it binomial?

• Finally, let’s consider a deck of cards, and we are interested in the number of hearts dealt in a hand of five. Call this number $X$. What is the pmf of $X$?

## $f(x)$ and $F(x)$

• $f(x)$ is the probability mass function of $X$. What does that mean? What is the connection to the distribution table? The probability histogram?

• $F(x)$ is the cumulative distribution function.

• What is the connection between $f$ and $F$?

# Concept Questions

01:00

Roll a pair of fair six-sided dice and let $X = 1$ if the dice land showing the same number of spots, and $0$ otherwise. For example, if both dice land $2$, then $X = 1$, but if one lands $2$ and the other lands $3$, then $X = 0$.

What is $P(X=1)$?

01:00

The graph of the cdf of a random variable $X$ is shown below. What is $F(2)$? What about $f(2)$?

01:00

You have $10$ people with a cold and you have a remedy with a $20\%$ chance of success. What is the chance that your remedy will cure at least one sufferer? (Let $X$ be the number of people cured among the 10. We are looking for the probability that $X \ge 1$)

What is the chance that at least one person is cured?

01:00

There are 4 histograms of different Poisson distributions below. Match each distribution to its parameter $\lambda$. Recall that $\lambda$ is how many occurrences we think will happen in a given period of time.

$(1)\: \lambda = 0.5 \hspace{2cm} (2)\: \lambda = 1 \hspace{2cm} (3) \: \lambda = 2 \hspace{2cm} (4) \: \lambda = 4\hspace{2cm}$

05:00

40:00