Random Variables

STAT 20: Introduction to Probability and Statistics

suggested times:

• last PS: time to work on it and then review (25 minutes)

• Brief lecture on random variables (20 minutes)

• Break (5 mins)

• Concept Questions (20 minutes)

• next PS (40 minutes)

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 \(P(X=1)\)?

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 \(6/36 = 1/6\)

01:00

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

\(F(2) = 2/3, f(2) = 0\)

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 \(X \sim Bin(10, p)\). \(1-p = 0.8 \: \& \: 1-0.8^{10} = 0.8926\)

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.

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

explain the smaller the rate, the lower the probability of higher values. A-4, B-1, C-3, D-2

Problem Set: Random Variables

20:00

Break

05:00

Lab 6

25:00