Random Variables

STAT 20: Introduction to Probability and Statistics

Agenda

• Brief lecture on probability distributions and random variables
• Review quiz 3
• Concept questions
• Break
• Handout: PS 4.1 (random variables)

Probability and quiz review

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?

Poll 1,000 Chicago residents and ask them if they voted for Lori Lightfoot in the mayoral election. Let \(X\) be the number of people who respond “Yes”. The population of Chicago is about 2.7 million.

Binomial, hypergeometric, or neither?

Hypergeometric(\(N = 2.7\) mill, \(n = 1000\), G), where G is not known, being the number of people who voted for Ms. Lightfoot. 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 4.1 (paper handout)