If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Course: Statistics and probability > Unit 9

Lesson 4: Combining random variables
Math
Statistics and probability
Random variables
Combining random variables
© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

Combining normal random variables

Google Classroom
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 50g and a standard deviation of 5g. 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 178g of candy.
Let's solve this problem by breaking it into smaller pieces.
Problem A (Example 1)
Find the mean of T.
μT=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
grams

Problem B (Example 1)
Find the standard deviation of T.
σT=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
grams

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

Problem D (Example 1)
Find the probability that a randomly selected bag contains less than 178g of candy.
Round to four decimal places.
P(T<178g)≈
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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=A−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.
μD=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
pins

Problem B (Example 2)
Find the standard deviation of D.
σD=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
pins

Problem C (Example 2)
What shape does the distribution of D have?
صرف 1 جواب چنو

Problem D (Example 2)
Find the probability that Mike scores higher than Adam in a randomly selected game.
Round to four decimal places.
P(Mike scores higher)≈
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
Hint: Find P(D<0).

Want to join the conversation?

Log in
No posts yet.
Do you understand English? Click here to see more discussion happening on Khan Academy's English site.