Discrete Random Variables
Instructor Katherine Cliff
Warm up question:
What is the expected value of a discrete random variable, and how do you calculate it?
Definition 1: Variance of a Discrete Random Variables
Example 1
For a raffle, a school sells 1,000 tickets for $2 each. There will be one prize for $1000. There will be three $100 prizes. There will be five $20 prizes. Compute the expected value and standard deviation for the value of a raffle ticket.
You work on numbers 2-4
2.
The probability distribution for the number of students in Statistics classes offered at a small college is given, but one value is missing. Fill in the missing value, then answer the questions that follow. Give your answer rounded to 4 decimal places when necessary.
| X | P(X) |
|---|---|
| 23 | 0.18 |
| 24 | 0.22 |
| 25 | 0.25 |
| 26 | |
| 27 | 0.2 |
mean
2.
The probability distribution for the number of students in Statistics classes offered at a small college is given, but one value is missing. Fill in the missing value, then answer the questions that follow. Give your answer rounded to 4 decimal places when necessary.
| X | P(X) |
|---|---|
| 23 | 0.18 |
| 24 | 0.22 |
| 25 | 0.25 |
| 26 | |
| 27 | 0.2 |
standard deviation
4.
X is a discrete random variable with probability mass function
\[P(X) = cx^2 \textrm{ for } x = \frac13, \frac23, 1, \frac43\]
A. Find the value of \(c\)
4.
X is a discrete random variable with probability mass function
\[P(X) = cx^2 \textrm{ for } x = \frac13, \frac23, 1, \frac43\]
B. Find the expected value of \(X\).
5.
Suppose that the probabilities are 0.2, 0.1, 0.5, and 0.2, respectively, that 0, 1, 2, or 3 snow storms will strike Colorado Springs in any given year. Find the mean and variance of the random variable X representing the number of snow storms in Colorado Springs in a given year.
Example 2
Let X be a discrete random variable with the following distribution:
| \(x_i\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \(P(X=x_i)\) | 0.1 | 0.4 | 0.3 | 0.2 |
Find \(E(X), E(2X-1), \text{Var}(2X-1)\), and \(E(X-2)^2\).
Example 2
Let X be a discrete random variable with the following distribution:
| \(x_i\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \(P(X=x_i)\) | 0.1 | 0.4 | 0.3 | 0.2 |
Find \(E(X), E(2X-1), \text{Var}(2X-1)\), and \(E(X-2)^2\).
Example 2
Let X be a discrete random variable with the following distribution:
| \(x_i\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \(P(X=x_i)\) | 0.1 | 0.4 | 0.3 | 0.2 |
Find \(E(X), E(2X-1), \text{Var}(2X-1)\), and \(E(X-2)^2\).
6.
Let X be a random variable with an expected value of E(X) = 28 and a variance Var(X) = 2. Find the expected value and variance of the following linear combinations:
A. E(X+4)
B. E(3X)
C. E(8X+2)
6.
Let X be a random variable with an expected value of E(X) = 28 and a variance Var(X) = 2. Find the expected value and variance of the following linear combinations:
D. Var(X+4)
E. Var(3X)
F. Var(8X+2)
Definition 2: Cumulative density function
Example 3
The cumulative distribution of a random variable X is given. Use it to find each of the following probabilities.
| \(x_i\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \(P(X \leq x_i)\) | 0.3 | 0.5 | 0.8 | 1 |
\[P(X \leq 2) = \]
\[P(X \geq 2) = \]
Example 3
The cumulative distribution of a random variable X is given. Use it to find each of the following probabilities.
| \(x_i\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \(P(X \leq x_i)\) | 0.3 | 0.5 | 0.8 | 1 |
\[P(X < 2) = \]
\[P(X > 2) = \]
7.
Use the cumulative probability distribution table to find the specified probabilities.
| \(x_i\) | \(P(X \leq x_i)\) |
|---|---|
| 0 | 0.12 |
| 1 | 0.27 |
| 2 | 0.43 |
| 3 | 0.83 |
| 4 | 1 |
\[P(X < 2)\]
\[P(X \leq 2)\]
7.
Use the cumulative probability distribution table to find the specified probabilities.
| \(x_i\) | \(P(X \leq x_i)\) |
|---|---|
| 0 | 0.12 |
| 1 | 0.27 |
| 2 | 0.43 |
| 3 | 0.83 |
| 4 | 1 |
\[P(X>1)\]
\[P(X \geq 1)\]
7.
Use the cumulative probability distribution table to find the specified probabilities.
| \(x_i\) | \(P(X \leq x_i)\) |
|---|---|
| 0 | 0.12 |
| 1 | 0.27 |
| 2 | 0.43 |
| 3 | 0.83 |
| 4 | 1 |
\[P(1 \leq X < 3)\]
8.
Let W be a random variable giving the number of heads that come up in four tosses of a coin.
A. Find the probability distribution of the random variable W. (Hint: Write out the sample space).
8.
Let W be a random variable giving the number of heads that come up in four tosses of a coin.
B. Find the expected value of the distribution.
C. What is the meaning of the expected value in this context?
8.
Let W be a random variable giving the number of heads that come up in four tosses of a coin.
A. Find the variance of the distribution.
