Random Variables

STAT 20: Introduction to Probability and Statistics

Agenda

• PS 8: time to work on it and then review
• Brief lecture and CQs on Random Variables
• Break
• PS 9 (random variables)
• If time, coin flipping graph

PS 8

25:00

Lecture

Random variables

• Let \(X\) be the number of heads in three tosses of a fair coin.

• 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 suppose we toss a fair coin until the first time it lands heads, and let \(X\) be the number of tosses. 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 \(X\)?

Write out outcome space of 3 coin flips of a fair coin and use this to define rv X = number of heads. Write out the pmf and cdf, and draw the cdf. Show why \(X\) is binomial and what are \(n\) ad \(p\)

Show what happens if coin is unfair. How the probabilities change.

Demo dbinom and pbinom and connection to pmf and cdf.

\(f(x)\) 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 \(F\)?

Draw an example on the board. Maybe box of tickets { 0, 0, 0 , 1, 1, 2, 2, 2} draw one ticket, and let \(X\) be the value of the ticket.

Concept Questions

01:00

Is the following random variable binomial (if so, what are \(n\) and \(p\)?), hypergeometric (if so , what are \(N\), \(G\), and \(n\)?), or neither?

Roll a fair ten-sided die 20 times. Let \(X\) be the number of times we roll a multiple of 3.

Binomial, hypergeometric, or neither?

Binomial(20, 3/10). The pollev will have three choices, and then figuring out the parameters can be a classroom discussion.

01:00

Is the following random variable binomial (if so, what are \(n\) and \(p\)?), hypergeometric (if so , what are \(N\), \(G\), and \(n\)?), or neither?

YouGov surveyed about 1,900 adults and asked them if they thought that Kevin Mcarthy should be ousted from his role as speaker. Let \(X\) be the number of people who responded “Yes”. The population of the US is about 335 million.

Binomial, hypergeometric, or neither?

Hypergeometric(\(N = 335\) mill, \(n = 1900\), G), where G is not known, being the number of people who approve of McCarthy’s ouster . Can remark that we could use Bin to model this rather than HG, and that we will discuss samples next week.

01:00

Is the following random variable binomial (if so, what are \(n\) and \(p\)?), hypergeometric (if so , what are \(N\), \(G\), and \(n\)?), or neither?

A six-sided die is tossed two times and the sum of the faces showing is \(8\). Let \(X\) be 1 if the sum is \(8\) and \(0\) otherwise.

Binomial, hypergeometric, or neither?

Binomial, with \(n = 1, p = 5/36\) or Bernoulli with \(p = 5/36\). This was part of the handout but I think some classroom discussion is warranted, so moved it here.

01:00

Is the following random variable binomial (if so, what are \(n\) and \(p\)?), hypergeometric (if so , what are \(N\), \(G\), and \(n\)?), or neither?

A bag that has 6 pieces of fruit: 2 mangoes, 3 apples, and 1 orange. I reach into the bag and draw out one fruit at a time, selecting each fruit at random (so they are equally likely). Let \(X\) be the number of draws until and including the first time I draw a apple.

Binomial, hypergeometric, or neither?

Neither. It would be useful at this point to write out the pmf of \(X\), and point out that they are using the multiplication rule and a conditional probability.

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?

\(P(X \ge 1) = 1 - P(X < 1) = 1 - P(X = 0) = 1 - (1-p)^{10}\), where \(p = 0.2\), because \(X \sim Bin(10, p)\). \(1-p = 0.8 \: \& \: 1-0.8^{10} = 0.8926\)

03:00

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 \(P(X=1)\)?

The chance that \(X = 1\) can be broken up into six mutually exclusive situations, that the dice both show one spot, or both show two spots, etc. Each of these has probability \(1/36\) so the total probability by the addition rule is \(6/36 = 1/6\)

Break

05:00

PS 9