Discrete Random Variables

We selected Q6.1.20 (p.307) as an example of using StatCrunch to build an expression for calculating the mean and standard deviation of a discrete random variable.

Q6.1.20 Waiting in Line A Wendy's manager performed a study to determine a probability distribution for the number of people, X , waiting in line during lunch. The results were as follows:

(a) Verify that this is a discrete probability distribution.

(b) Draw a probability histogram.

(d) Compute the standard deviation of the random variable .

(e) What is the probability that eight people are waiting in line for lunch?

(f) What is the probability that 10 or more people are waiting in line for lunch? Would this be unusual?

(a) Verify that this is a discrete probability distribution.

This is a discrete probability distribution because each Px() is between 0 and 1 and the sum of Px() is 1  Px( ) 1 .

(b) Draw a probability histogram.

Step 1: 1) Download the data. 2) Click Stat → Calculators → Custom.

Step 2: 1) Choose X for Values in:, and Px() for Weights in:. 2) Click Compute!

Step 3: StatCrunch output of the probability histogram is shown to the right.

Use the formula X [x P ( x )] to build an expression for calculating the mean of the discrete random variable.

Step 1: Click Data → Compute Expression.

Step 2: In the Compute Expression dialogue box, click Build under Expression:

The Expression dialogue box will pop up. In the empty space bar where the red cursor is located, follow the instructions in Step 3 and Step 4 to enter the formula [x P ( x )] .

Step 3: Click sum under Functions: then click Add Function.

Step 4: Click x under Columns: then click Add Column. Click * on the calculator template. Click Px() under Columns: then click Add Column. Click Okay.

Step 5: Click Compute!

The mean of the discrete random variable is computed and displayed in the column next to Px()as shown below.

Round your answer to one decimal place --> X = 4.9 . Interpretation: In the long run, during lunch time the average number of people waiting in line in Wendy is 4.9.

(d) Compute the standard deviation of the random variable X .

22 Use the formulaXX[x  P ( x )]  to build an expression for calculating the standard deviation of the discrete random variable.

Step 1: Click Data → Compute Expression.

Step 2: In the Compute Expression dialogue box, click Build under Expression:

The Expression dialogue box will pop up. In the empty space bar where the red cursor is located, follow the instructions in Step 3 and Step 4 to enter the formula

22 XX[x  P ( x )]  .

Step 3: Click sqrt under Functions: → Add Function.

Step 4: Under Functions:, click sum → Add Function.

Step 5: Click x under Columns: → Add Column → ^ → 2 → * Click Px() under Columns: → Add Column → Move the cursor one space to the right so the red cursor is on the last closing parenthesis.

Step 6: Click – Click ( ) Click sum(x * P(x)) under Columns: → Add Column → Move the cursor one space to the right so the red cursor is in between two closing parentheses. Click ^ then click 2 . Click Okay.

Step 7: When the Compute Expression box pops up, click Compute!

The standard deviation, X , of the discrete random variable is computed and displayed on the column next to sum(x*P(x)).

(e) What is the probability that eight people are waiting in line for lunch?

From the table given,

the probably that eight people are waiting in line for lunch is Px( 8) = 0.063.

(f) What is the probability that 10 or more people are waiting in line for lunch? Would this be unusual?

From the table given,

P( x 10) P ( x 10) P ( x 11) P ( x 12) 0.019  0.004  0.006  0.029 Since Px( 10) is 0.029 which is less than 0.05, it is unusual to have 10 or more people waiting in line for lunch.