section 7.2. mean of a probability distribution is the long- run average outcome, µ, or µ x. also...

Section 7.2

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Section 7.2

Mean of a probability distribution is the long-run average outcome, µ, or µx.

Also called the expected value of x, or E(X).

µx = ∑ xi Pi

Example: X heads in four coin tosses.Find the mean of X.X i 0 1 2 3 4

Pi .0625 .25 .375 .25 .0625

Page 3: Section 7.2. Mean of a probability distribution is the long- run average outcome, µ, or µ x. Also called the expected value of x, or E(X). µ x = x i P

Page 4: Section 7.2. Mean of a probability distribution is the long- run average outcome, µ, or µ x. Also called the expected value of x, or E(X). µ x = x i P

Inhabitants: 1 2 3 4 5 6 7Proportion: .25 .32 .17 .15 .07 .03 .01Find mean household size.

Page 5: Section 7.2. Mean of a probability distribution is the long- run average outcome, µ, or µ x. Also called the expected value of x, or E(X). µ x = x i P

In a sample that uses mean x to estimate the value of the population mean µ, as the # of observations increases, mean x eventually approaches mean µ.

Page 6: Section 7.2. Mean of a probability distribution is the long- run average outcome, µ, or µ x. Also called the expected value of x, or E(X). µ x = x i P

1. If X is a random variable and a and b are fixed numbers, then: µa+bx = a + bµx

2. If X and Y are random variables, then: µX+Y = µX + µY

Example: Dan (X) and Danna (Y) are on a math team. Each takes 10 tests on a variety of concepts. Their scores are then combined and averaged for the team score.

µX = 82 µY = 86 Find the mean score of the team, µX+Y

Page 7: Section 7.2. Mean of a probability distribution is the long- run average outcome, µ, or µ x. Also called the expected value of x, or E(X). µ x = x i P

σx2

Variance of random variable X: σx

2 = ∑(xi - µX)2 pi

Example: Given the following distribution, find the mean and standard deviation:

Units Sold 1000 3000 5000 10,000

Probability .1 .3 .4 .2

Page 8: Section 7.2. Mean of a probability distribution is the long- run average outcome, µ, or µ x. Also called the expected value of x, or E(X). µ x = x i P

1. If X is a random variable and a and b are fixed numbers, then σ2

a+bx = b2σx2

2. If X and Y are independent random variables, then σ2

x+y = σx2 + σy

σ2x-y = σx

2 + σy2

3. What about standard deviation?

Page 9: Section 7.2. Mean of a probability distribution is the long- run average outcome, µ, or µ x. Also called the expected value of x, or E(X). µ x = x i P

Two golf players, Jake(X) and Carrine(Y), have the following stats:

µx = 110, σx = 10

µy = 100, σy = 8

Find the difference in their mean scores.Find the variance of the difference between

their scores.Find the standard deviation of the difference

in their scores.

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