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# Combining normal random variables

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

## Example 1: Total amount of candy

Each bag of candy is filled at a factory by 4 machines. The first machine fills the bag with blue candies, the second with green candies, the third with red candies, and the fourth with yellow candies. The amount of candy each machine dispenses is normally distributed with a mean of 50, start text, g, end text and a standard deviation of 5, start text, g, end text. Also, assume that the amount dispensed by any given machine is independent from the other machines.
Let T be the total weight of candy in a randomly selected bag.
Find the probability that a randomly selected bag contains less than 178, start text, g, end text of candy.
Let's solve this problem by breaking it into smaller pieces.
Problem A (Example 1)
Find the mean of T.
mu, start subscript, T, end subscript, equals
grams

Problem B (Example 1)
Find the standard deviation of T.
sigma, start subscript, T, end subscript, equals
grams

Problem C (Example 1)
What shape does the distribution of T have?

Problem D (Example 1)
Find the probability that a randomly selected bag contains less than 178, start text, g, end text of candy.
Round to four decimal places.
P, left parenthesis, T, is less than, 178, start text, g, end text, right parenthesis, approximately equals

## Example 2: Difference in bowling scores

Adam and Mike go bowling every week. Adam's scores are normally distributed with a mean of 175 pins and a standard deviation of 30 pins. Mike's scores are normally distributed with a mean of 150 pins and a standard deviation of 40 pins. Assume that their scores in any given game are independent.
Let A be Adam's score in a random game, M be Mike's score in a random game, and D be the difference between Adam's and Mike's scores where D, equals, A, minus, M.
Find the probability that Mike scores higher than Adam in a randomly selected game.
Let's solve this problem by breaking it into smaller pieces.
Problem A (Example 2)
Find the mean of D.
mu, start subscript, D, end subscript, equals
pins

Problem B (Example 2)
Find the standard deviation of D.
sigma, start subscript, D, end subscript, equals
pins

Problem C (Example 2)
What shape does the distribution of D have?