Combining Normal Random Variables

Combining normal random variables
When we combine random variables that each follow a normal distribution, the resulting distribution is also normally distributed. This lets us answer questions about the resulting distribution.

Example Problems: Combining Normal Random Variables

Example 1

Andy, Bob, Craig, and Doug play a round of golf together each week. Each of their scores are normally distributed with a mean of 70 strokes and a standard deviation of 5 strokes. Find the probability that the sum of their scores in any given week is higher than 300 strokes.

Step 1: Find the mean number of total strokes.

μtotal = μAndy + μBob + μCraig + μDoug

μtotal = 70 + 70 + 70 +70 = 280

Step 2: Find the standard deviation of total strokes.

σ2total = σ2Andy + σ2Bob + σ2Craig + σ2Doug

σ2total = 52 + 52 + 52 + 52 = 100

σtotal = √100 = 10

Step 3: Use the Z Score Area Calculator to find the probability that the sum of their scores in a given week is higher than 300 strokes. 

Combining normal random variables
The probability that the sum of their scores in a given week is above 300 is 0.02275.

Example 2

Meredith and Angela are both saleswomen at a paper company. Meredith’s weekly sales are normally distributed with a mean of 50 sales and a standard deviation of 2 sales. Angela’s weekly sales are normally distributed with a mean of 45 sales and a standard deviation of 2 sales.

Assume that their sales in any given week are independent. Find the probability that Angela has more sales than Meredith in a randomly selected week.

Step 1: Find the mean of the difference in sales.

μdiff = μMeredith – μAngela

μdiff = 50 – 45 = 5

Step 2: Find the standard deviation of the difference in sales.

σ2diff = σ2Meredith + σ2Angela

σ2diff = 22 + 22 = 4

σdiff = √4 = 2

Step 3: Use the Z Score Area Calculator to find the probability that the difference between Meredith and Angela’s sales is less than zero in a given week.

Normal random variables probability
The probability that the difference between Meredith and Angela’s sales is less than zero in a given week. (i.e. Angela sells more) is 0.00621.

Example 3

Mike, Greg, Austin, and Tony play a game of bowling each week together. Each of their scores are normally distributed with a mean of 100 and a standard deviation of 5. Find the probability that the sum of their scores in any given week is lower than 415.

Step 1: Find the mean number of total strokes.

μtotal = μMike + μGreg + μAustin + μTony

μtotal = 100 + 100 + 100 + 100 = 400

Step 2: Find the standard deviation of total strokes.

σ2total = σ2Mike + σ2Greg + σ2Austin + σ2Tony

σ2total = 52 + 52 + 52 + 52 = 100

σtotal = √100 = 10

Step 3: Use the Z Score Area Calculator to find the probability that the sum of their scores in a given week is lower than 415. 


The probability that the sum of their scores in a given week is below 415 is 0.93319.

Example 4

Cody and Kyle are both salesmen at a sneaker company. Cody’s weekly sales are normally distributed with a mean of 30 sales and a standard deviation of 2 sales. Kyle’s weekly sales are normally distributed with a mean of 25 sales and a standard deviation of 2 sales.

Assume that their sales in any given week are independent. Find the probability that Kyle has more sales than Cody in a randomly selected week.

Step 1: Find the mean of the difference in sales.

μdiff = μCody – μKyle

μdiff = 30 – 25 = 5

Step 2: Find the standard deviation of the difference in sales.

σ2diff = σ2Cody + σ2Kyle

σ2diff = 22 + 22 = 4

σdiff = √4 = 2

Step 3: Use the Z Score Area Calculator to find the probability that the difference between Cody and Kyle’s sales is less than zero in a given week.

Normal random variables probability example
The probability that the difference between Cody and Kyle’s sales is less than zero in a given week. (i.e. Kyle sells more) is 0.00621.

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