Mean and Mean and Standard Standard DeviationDeviation

Lecture 23Lecture 23

Section 7.5.1Section 7.5.1

Mon, Oct 17, 2005Mon, Oct 17, 2005

The Mean and Standard The Mean and Standard DeviationDeviation

Mean of a Discrete Random VariableMean of a Discrete Random Variable – The average of the values that the – The average of the values that the random variable takes on, in the random variable takes on, in the long run.long run.

Standard Deviation of a Discrete Standard Deviation of a Discrete Random VariableRandom Variable – The standard – The standard deviation of the values that the deviation of the values that the random variable takes on, in the random variable takes on, in the long run.long run.

The Mean of a Discrete The Mean of a Discrete Random VariableRandom Variable

The mean is also called the The mean is also called the expected expected valuevalue..

However, that does not mean that it However, that does not mean that it is is literallyliterally the value that we expect the value that we expect to see.to see.

““Expected value” is simply a Expected value” is simply a synonym for the mean or average.synonym for the mean or average.

The Mean of a Discrete The Mean of a Discrete Random VariableRandom Variable

The mean, or expected value, of The mean, or expected value, of XX may be denoted by either of two may be denoted by either of two symbols.symbols.

µ or µ or EE((XX)) If another random variable is called If another random variable is called YY, then we would write , then we would write EE((YY).).

Or we could write them as µOr we could write them as µXX and µ and µYY..

Computing the MeanComputing the Mean

Given the pdf of Given the pdf of XX, the mean is , the mean is computed ascomputed as

This is a weighted average of This is a weighted average of XX.. Each value is weighted by its likelihood.Each value is weighted by its likelihood.

xXPx

xXPxxXPx nn

11

Example of the MeanExample of the Mean

Recall the example where Recall the example where XX was the was the number of children in a household.number of children in a household.

x P(X = x)

00 0.100.10

11 0.300.30

22 0.400.40

33 0.200.20

Example of the MeanExample of the Mean

x P(X = x) xP(X = x)

00 0.100.10 0.000.00

11 0.300.30 0.300.30

22 0.400.40 0.800.80

33 0.200.20 0.600.60

Multiply each Multiply each xx by the corresponding by the corresponding probability.probability.

Example of the MeanExample of the Mean

x P(X = x) xP(X = x)

00 0.100.10 0.000.00

11 0.300.30 0.300.30

22 0.400.40 0.800.80

33 0.200.20 0.600.601.70 = µ

Add up the column of products to get Add up the column of products to get the mean.the mean.

Let’s Do It!Let’s Do It!

Let’s do it! 7.23, p. 430 – Profits and Let’s do it! 7.23, p. 430 – Profits and Weather.Weather.

The Variance of a The Variance of a Discrete Random Discrete Random

VariableVariable Variance of a Discrete Random Variance of a Discrete Random

VariableVariable – The average squared – The average squared deviation of the values that the deviation of the values that the random variable takes on, in the long random variable takes on, in the long run.run.

The variance of The variance of XX is denoted by is denoted by

22 or Var( or Var(XX)) The standard deviation of The standard deviation of XX is is

denoted by denoted by ..

The Variance and The Variance and Expected ValuesExpected Values

The variance is the expected value of The variance is the expected value of the squared deviations.the squared deviations.

That agrees with the earlier notion That agrees with the earlier notion of the average squared deviation.of the average squared deviation.

Therefore,Therefore,

2 XEXVar

Example of the VarianceExample of the Variance

x P(X = x)

00 0.100.10

11 0.300.30

22 0.400.40

33 0.200.20

Again, let Again, let XX be the number of be the number of children in a household.children in a household.

Example of the VarianceExample of the Variance

x P(X = x) x – µ

00 0.100.10 -1.7-1.7

11 0.300.30 -0.7-0.7

22 0.400.40 +0.3+0.3

33 0.200.20 +1.3+1.3

Subtract the mean (1.70) from each Subtract the mean (1.70) from each value of X to get the deviations.value of X to get the deviations.

Example of the VarianceExample of the Variance

x P(X = x) x – µ (x – µ)2

00 0.100.10 -1.7-1.7 2.892.89

11 0.300.30 -0.7-0.7 0.490.49

22 0.400.40 +0.3+0.3 0.090.09

33 0.200.20 +1.3+1.3 1.691.69

Square the deviations.Square the deviations.

Example of the VarianceExample of the Variance

x P(X = x) x – µ (x – µ)2 (x – µ)2P(X = x)

00 0.100.10 -1.7-1.7 2.892.89 0.2890.289

11 0.300.30 -0.7-0.7 0.490.49 0.1470.147

22 0.400.40 +0.3+0.3 0.090.09 0.0360.036

33 0.200.20 +1.3+1.3 1.691.69 0.3380.338

Multiply each squared deviation by its Multiply each squared deviation by its probability.probability.

Example of the VarianceExample of the Variance

x P(X = x) x – µ (x – µ)2 (x – µ)2P(X = x)

00 0.100.10 -1.7-1.7 2.892.89 0.2890.289

11 0.300.30 -0.7-0.7 0.490.49 0.1470.147

22 0.400.40 +0.3+0.3 0.090.09 0.0360.036

33 0.200.20 +1.3+1.3 1.691.69 0.3380.338

0.810 = 2

Add up the products to get the Add up the products to get the variance.variance.

Example of the VarianceExample of the Variance

x P(X = x) x – µ (x – µ)2 (x – µ)2P(X = x)

00 0.100.10 -1.7-1.7 2.892.89 0.2890.289

11 0.300.30 -0.7-0.7 0.490.49 0.1470.147

22 0.400.40 +0.3+0.3 0.090.09 0.0360.036

33 0.200.20 +1.3+1.3 1.691.69 0.3380.338

0.810 = 2

0.9 =

Add up the products to get the Add up the products to get the variance.variance.

Alternate Formula for Alternate Formula for the Variancethe Variance

It turns out thatIt turns out that

That is, the variance of That is, the variance of XX is “the is “the expected value of the square of expected value of the square of XX minus the square of the expected minus the square of the expected value of value of XX.”.”

Of course, we could write this asOf course, we could write this as

22 XEXEXVar

22 XEXVar

Example of the VarianceExample of the Variance

x P(X = x)

00 0.100.10

11 0.300.30

22 0.400.40

33 0.200.20

One more time, let One more time, let XX be the number be the number of children in a household.of children in a household.

Example of the VarianceExample of the Variance

x P(X = x) x2

00 0.100.10 00

11 0.300.30 11

22 0.400.40 44

33 0.200.20 99

Square each value of Square each value of XX..

Example of the VarianceExample of the Variance

x P(X = x) x2 x2P(X = x)

00 0.100.10 00 0.000.00

11 0.300.30 11 0.300.30

22 0.400.40 44 1.601.60

33 0.200.20 99 1.801.80

Multiply each squared Multiply each squared XX by its by its probability.probability.

Example of the VarianceExample of the Variance

x P(X = x) x2 x2P(X = x)

00 0.100.10 00 0.000.00

11 0.300.30 11 0.300.30

22 0.400.40 44 1.601.60

33 0.200.20 99 1.801.80

3.70 = E(X2)

Add up the products to get Add up the products to get EE((XX22).).

Example of the VarianceExample of the Variance

Then use Then use EE((XX22) and µ to compute the ) and µ to compute the variance.variance.

Var(Var(XX) = ) = EE((XX22) – µ) – µ22

= 3.70 – (1.7)= 3.70 – (1.7)22

= 3.70 – 2.89= 3.70 – 2.89

= 0.81.= 0.81. It follows that It follows that = = 0.81 = 0.9.0.81 = 0.9.

TI-83 – Means and TI-83 – Means and Standard DeviationsStandard Deviations

Store the list of values of Store the list of values of XX in L in L11..

Store the list of probabilities of Store the list of probabilities of XX in L in L22.. Select STAT > CALC > 1-Var Stats.Select STAT > CALC > 1-Var Stats. Press ENTER.Press ENTER. Enter LEnter L11, L, L22.. Press ENTER.Press ENTER. The list of statistics includes the mean and The list of statistics includes the mean and

standard deviation of X.standard deviation of X. Use Use x, not Sx, for the standard deviation.x, not Sx, for the standard deviation.

TI-83 – Means and TI-83 – Means and Standard DeviationsStandard Deviations

Let LLet L11 = {0, 1, 2, 3}. = {0, 1, 2, 3}.

Let LLet L22 = {0.1, 0.3, 0.4, 0.2}. = {0.1, 0.3, 0.4, 0.2}. Compute the statistics.Compute the statistics. Compute µ and Compute µ and for the Indoor and for the Indoor and

Outdoor distributions in Let’s Do It! Outdoor distributions in Let’s Do It! 7.23, p. 430.7.23, p. 430.

Let’s Do It!Let’s Do It!

Return once more to Let’s Do It! 7.23, Return once more to Let’s Do It! 7.23, p. 430.p. 430. The standard deviation of Profit Outdoors The standard deviation of Profit Outdoors

is 23.9.is 23.9. Use the Use the originaloriginal formula to compute the formula to compute the

standard deviation of Profit Indoors.standard deviation of Profit Indoors. Use the Use the alternatealternate formula to compute the formula to compute the

standard deviation of Profit Indoors.standard deviation of Profit Indoors. Use the TI-83 to find the standard Use the TI-83 to find the standard

deviation of Profit Indoors.deviation of Profit Indoors.