Random Variables

STAT 20: Introduction to Probability and Statistics

suggested times:

• last worksheet: 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 worksheet (40 minutes)

Warmup

15:00

Tossing a fair coin 3 times and counting the number of heads

Let \(X\) be the number of heads in three tosses of a fair coin. What is the probability mass function \(f(x) = P(X = x)\) of \(X\)? What is this distribution called? What values will \(x\) take?

• \(f(x) = P(X = x)\), where \(X\) has the Binomial distribution
• \(n=3\) and \(p=\dfrac{1}{2}\).
• \(f(x) = \displaystyle \binom{3}{x}\left(\dfrac{1}{2}\right)^x \left(\dfrac{1}{2}\right)^{3-x}\)
• \(x\) takes the values \(0, 1, 2, 3\)

Tossing a biased coin 3 times and counting the number of heads

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 \(f(x) = P(X = x)\)?

• \(f(x) = P(X = x)\), where \(X\) has the Binomial distribution
• \(n=3\) and \(p=\dfrac{2}{3}\).
• \(f(x) = \displaystyle \binom{3}{x}\left(\dfrac{2}{3}\right)^x \left(\dfrac{1}{3}\right)^{3-x}\)
• \(x\) takes the values \(0, 1, 2, 3\)

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 \(n\) and \(p\)

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

Optional: Demo dbinom and pbinom and connection to pmf and cdf. Show how we can use rbinom to get an empirical histogram for three flips of fair or biased coin.

Tossing a fair coin until it lands heads

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 probability mass function of \(X\)? Is it binomial?

\(P(X = x) = f(x) = \left(\dfrac{1}{2}\right)^{x-1}\left(\dfrac{1}{2}\right)\).

Not binomial since \(n\) not specified at the start.

What are the possible values of \(x\)?

Here, it is a good idea to do it slowly so that they understand what is different about this distribution. You can switch it to \(P(H) = p\) and ask what would be the probability etc.

Dealing cards and counting hearts

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 \(X\)?

Let \(X\) be the number of hearts in the hand of 5 cards. We know that \(X\) takes the values \(0, 1, 2,\ldots, 5\). Since we deal cards without replacement, we see that \(X\) has the hypergeometric distribution with parameters \(N = 52, \, G = 13, \, n =5\).

Concept review: \(f(x)\) 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. Write out both \(f(x)\) and \(F(x)\) in function notation, then draw the cdf and show how we can get from the cdf to the pmf.

Concept Questions

Rolling dice

Roll a pair of fair six-sided dice. Let \(X = 1\) if the dice show the same number of spots, and \(0\) otherwise.

For example, if both dice show \(2\), then \(X = 1\), but if one shows \(2\) and the other shows \(3\), then \(X = 0\).

What is \(P(X=1)\)?

01:00

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 \(6/36 = 1/6\)

Graphing the CDF

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

01:00

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

There’s a cold going around…

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

Poisson distributions

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

01:00

Worksheet: Random Variables

20:00