1. 7.2 Means & Variances of Random Variables AP Statistics

2. Mean Definition: The mean of a random variable (X) is an average of the possible values of the random variable with accommodation for the fact that not all outcomes are necessarily equally likely. The mean of a random variable is a “weighted average”. The mean of a random variable is often called the expected value of that variable. To find the mean (μ) of a discrete random variable: sum the products of each possible value and its probability.

3. Example 1: Consider the following distribution of grades on an AP Statistics Exam Grade 0 1 2 3 4 Probability0.10 0.15 0.30 0.30 0.15 Find the average grade on this test. µ x = 0(.10) + 1(.15) + 2(.30) + 3(.30) + 4(.15) = 2.25

4. To find the mean of a continuous random variable, would require us to find the “balance point” of the density curve. Unless the curve is symmetric, in which case the mean would be at the center, finding the mean requires mathematics beyond the scope of this course.

5. Variance Remember that variance and standard deviation are measures of spread of a distribution. The variance is the average of the squared differences of a variable from its mean. The standard deviation is the square root of the variance. To find the variance of a discrete random variable: multiply the squared difference of each value from the mean by the probability for that particular value.

6. Example: The variance of the grade distribution from the AP Statistics Exam would be: σ x 2 = (0 – 2.25) 2 (.10) + (1 – 2.25) 2 (.15) + (2 – 2.25) 2 (.30) + (3 – 2.25) 2 (.30) + (4 – 2.25) 2 (.15) σ x 2 = 1.3875 The standard deviation would be: σ x = = 1.178

7. Calculating the Mean and Variance of a Discrete Random Variable Calculate the mean for the random variable X: µ = __7.16____ X012345678910 P(X).002.001.002.005.02.04.17.38.25.12.01

8. Use the following table to calculate the standard deviation of X: XP(X) 0.002 1.001 2.002 3.005 4.02 5.04 6.17 7.38 8.25 9.12 10.01 = σ x = 1.252.1025.0379.0533.0865.1997.1866.2288.0097.1764.4063.0801 1.5678 L1L1 L2L2 L 3 =(L 1 -μ) 2 *L 3

9. Rules for Means and Variances 1. If a number is multiplied by every value of a random variable, the mean of that random variable is multiplied by the same factor. If a number is added to every value of a random variable, the mean is increased by that same value.

10. 2. If a new random variable is created by adding (or subtracting) two random variables, then the mean of the new random variable is the sum (or difference) of the means of the original variables.

11. 3. If a number is multiplied by every value of a random variable, the variance of that random variable is multiplied by the square of the number. If a number is added to every value of a random variable, then the variance of the random variable is unchanged.

12. 4. If a new random variable is created by adding (or subtracting) two random variables, then the variance of the new random variable is the sum of the variances of the original variable.

13. Examples for Rules of Means & Variances

14. 1. Let L = 3 + 2X. Find mean, variance and standard deviation of L.

15. 2. Let W = X + Y. Find mean, variance and standard deviation of W.

16. 3. Let W = X - Y. Find mean, variance and standard deviation of W.