7.2 Means and Variances of Random Variables

1. 7.2 Means and Variances of Random Variables

2. Review

3. Review

4. Review

5. Review

6. Mean of a Discrete Random Variable The mean value of a discrete random variable x, denoted by mx, is computed by first multiplying each possible x value by the probability of observing that value and then adding the resulting quantities. Symbolically,

7. Example • A professor regularly gives multiple choice quizzes with 5 questions. Over time, he has found the distribution of the number of wrong answers on his quizzes is as follows

8. Example • Multiply each x value by its probability and add the results to get mx. mx = 1.41

10. Variance and Standard Deviation of a Discrete Random Variable

11. Previous Example - continued

15. The Mean & Variance of a Linear Function

16. Example Suppose x is the number of sales staff needed on a given day. If the cost of doing business on a day involves fixed costs of $255 and the cost per sales person per day is $110, find the mean cost (the mean of x or mx) of doing business on a given day where the distribution of x is given below.

17. Example continued We need to find the mean of y = 255 + 110x

18. Example continued We need to find the variance and standard deviation of y = 255 + 110x

19. Means and Variances for Linear Combinations If x1, x2,  , xn are random variables and a1, a2,  , an are numerical constants, the random variable y defined as y = a1x1 + a2x2 + + anxn is a linear combination of the xi’s.

22. If x1, x2,  , xn are random variables with means m1, m2,  , mn and variances respectively, and y = a1x1 + a2x2 + + anxn then 2. If x1, x2,  , xn are independent random variables then and Means and Variances for Linear Combinations 1. my = a1m1 + a2m2 + + anmn (This is true for any random variables with no conditions.)

23. Apples Oranges Grapes Mean 8 10 7 m Standard deviation 0.9 1.1 2 s Example A distributor of fruit baskets is going to put 4 apples, 6 oranges and 2 bunches of grapes in his small gift basket. The weights, in ounces, of these items are the random variables x1, x2 and x3 respectively with means and standard deviations as given in the following table. Find the mean, variance and standard deviation of the random variable y = weight of fruit in a small gift basket.

24. Apples Oranges Grapes Mean 8 10 7 m Standard deviation 0.9 1.1 2 s Example continued It is reasonable in this case to assume that the weights of the different types of fruit are independent.

25. Summary • The Law of Large Numbers says that the average of the values of X observed in many trials approach μ. • If X is discrete with the possible values xi, the means is the average of the values of X, each weighted by its probability: μX = x1p1 + x2p2 + … + xnpn • The variance σ2 for a discrete variable: σ2X = (x1 - μ)2p1 + (x2 – μ)2p2 + … + (xn - μ)2pn • The standard deviation σX is the square root of the variance.

26. Summary The means and variance of random variables obey the following rules • If a nd b are fixed numbers, then μa + bX = a + b μX σ2a+bX = b2σ2X • If X and Y are any two random variables, then μX + Y = μX + μY • And if X and Y are independent, then σ2X + Y = σ2X + σ2Y σ2X – Y = σ2X + σ2Y