AP Statistics Section 7.2 C

Rules for Means & Variances

Consider the independent random variables X and Y and their probability distributions below:

41.2)3(.)7.25()5(.)7.22()2(.)7.21( 2222 x

7.2)3(.5)5(.2)2(.1 x

7.2 41.2 6.2 84.

Build a new random variable X + Y and calculate the probabilities for the values of X + Y.

3

9 7

6 4

5 9 7 6 5 4 3

.14.7.22)P(1P(3)

.09 .21 .15 .06 .35 .14

Use your calculator to calculate the mean of the random variable X + Y.

Note that the mean of the sum = ____ equals the sum of the means =______________ :

3.5yx

yxyx

3.53.56.27.2

Use your calculator to calculate the variance of the random variable X + Y.

Note that the variance of the sum equals the sum of the variances:

25.32 yx

25.384.41.222 yx

Repeat the steps above for the random variable X – Y.

Verify .

1 3

2- 0

3- 1

3 1 0 1 2 3 .21 .09 .35 .14 .15 .06

yxyx

.1 .1

2.6 2.7 1.

Repeat the steps above for the random variable X – Y.

Calculate the variance of the random variable X – Y.

Note that the variance of the difference equals

1 3

2- 0

3- 1

3 1 0 1 2 3 .21 .09 .35 .14 .15 .06

25.32 yx

2yx

2y

2x and variances theof sum the

Rules for MeansRule 1: If X is a random variable and a and b are

constants, then ._______bxa

well.asmean the toadded is then x,of each value toadded is If aa

well.as by multiplied

ismean then the,by multiplied is x of each value If

b

b

xba

Rules for Means

Rule 2: If X and Y are random variables, then ______and _______yx yxyx yx

Rules for VariancesRule 1: If X is a random variable and a and b are

constants, then ______2bxa

variance. thechangenot does x of each value to Adding a

2by variance themultiplies ,by x of each value gMultiplyin bb

22xb

Rules for VariancesRule 2: If X and Y are independent random

variables, then222

y-x222 and yxyxyx

Example: Consider two scales in a chemistry lab. Both scales give answers that vary a little in repeated weighings of the same item. For a 2 gram item, the first scale gives readings X with a mean of 2g and a standard deviation of .002g. The second scale’s readings Y have a mean of 2.001g and a standard deviation of .001g.

If X and Y are independent, find the mean and standard deviation of Y – X.

gxyxy 001.2001.2

000005.001.002. 22222 xyxy

002236.000005. xy

Example: Consider two scales in a chemistry lab. Both scales give answers that vary a little in repeated weighings of the same item. For a 2 gram item, the first scale gives readings X with a mean of 2g and a standard deviation of .002g. The second scale’s readings Y have a mean of 2.001g and a standard deviation of .001g.

You measure once with each scale and average the readings. Your result is Z = (X+Y)/2. Find .

0005.2)001.2(5.)2(5.21

21 yxz

yx 21

21z :Note

00000125.)001(.21)002(.2

12

12

1 2222222

21

21

2 yxyxz

001118034.00000125. z

Any linear combination of independent Normal random

variables is also Normally distributed.

Example: Tom and George are playing in the club golf tournament. Their scores vary as they play the course repeatedly. Tom’s score X has the N(110, 10) distribution and George’s score Y has the N(100, 8) distribution. If they play independently, what is the probability that Tom will score lower than George?

)0()( YXPYXP

806.12810

10100110

22

yx

yx

0 806.12

10

2177.

78.12.806

10-0z

:Table

.2174

0,12.806)-10000,0,1normalcdf(

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