Mean and Standard Deviation Lecture 24 Section 7.5.1 Wed, Oct 27, 2004

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

The Mean of a Discrete Random Variable The mean is also called the expected value. However, that does not mean that it is literally the value that we expect to see. “Expected value” is simply a synonym for the mean or average.

The Mean of a Discrete Random Variable The mean, or expected value, of X may be denoted by either of two symbols. µ or E(X) If another random variable is called Y, then we would write E(Y). Or we could write them as µX and µY.

Computing the Mean Given the pdf of X, the mean is computed as This is a weighted average of X. Each value is weighted by its likelihood.

Example of the Mean Recall the example where X was the number of children in a household. x P(X = x) 0.10 1 0.30 2 0.40 3 0.20

Example of the Mean Multiply each x by the corresponding probability. P(X = x) xP(X = x) 0.10 0.00 1 0.30 2 0.40 0.80 3 0.20 0.60

Example of the Mean Add up the column of products to get the mean. x P(X = x) xP(X = x) 0.10 0.00 1 0.30 2 0.40 0.80 3 0.20 0.60 1.70 = µ

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

The Variance of a Discrete Random Variable Variance of a Discrete Random Variable – The square of the standard deviation of that random variable. The variance of X is denoted by 2 or Var(X) The standard deviation of X is denoted by .

The Variance and Expected Values The variance is the expected value of the squared deviations. That agrees with the earlier notion of the average squared deviation. Therefore,

Example of the Variance Again, let X be the number of children in a household. x P(X = x) 0.10 1 0.30 2 0.40 3 0.20

Example of the Variance Subtract the mean (1.70) from each value of X to get the deviations. x P(X = x) x – µ 0.10 -1.7 1 0.30 -0.7 2 0.40 +0.3 3 0.20 +1.3

Example of the Variance Square the deviations. x P(X = x) x – µ (x – µ)2 0.10 -1.7 2.89 1 0.30 -0.7 0.49 2 0.40 +0.3 0.09 3 0.20 +1.3 1.69

Example of the Variance Multiply each squared deviation by its probability. x P(X = x) x – µ (x – µ)2 (x – µ)2P(X = x) 0.10 -1.7 2.89 0.289 1 0.30 -0.7 0.49 0.147 2 0.40 +0.3 0.09 0.036 3 0.20 +1.3 1.69 0.338

Example of the Variance Add up the products to get the variance. x P(X = x) x – µ (x – µ)2 (x – µ)2P(X = x) 0.10 -1.7 2.89 0.289 1 0.30 -0.7 0.49 0.147 2 0.40 +0.3 0.09 0.036 3 0.20 +1.3 1.69 0.338 0.810 = 2

Example of the Variance Add up the products to get the variance. x P(X = x) x – µ (x – µ)2 (x – µ)2P(X = x) 0.10 -1.7 2.89 0.289 1 0.30 -0.7 0.49 0.147 2 0.40 +0.3 0.09 0.036 3 0.20 +1.3 1.69 0.338 0.810 = 2 0.9 = 

Alternate Formula for the Variance It turns out that That is, the variance of X is “the expected value of the square of X minus the square of the expected value of X.” Of course, we could write this as

Example of the Variance One more time, let X be the number of children in a household. x P(X = x) 0.10 1 0.30 2 0.40 3 0.20

Example of the Variance Square each value of X. x P(X = x) x2 0.10 1 0.30 2 0.40 4 3 0.20 9

Example of the Variance Multiply each squared X by its probability. x P(X = x) x2 x2P(X = x) 0.10 0.00 1 0.30 2 0.40 4 1.60 3 0.20 9 1.80

Example of the Variance Add up the products to get E(X2). x P(X = x) x2 x2P(X = x) 0.10 0.00 1 0.30 2 0.40 4 1.60 3 0.20 9 1.80 3.70 = E(X2)

Example of the Variance Then use E(X2) to compute the variance. Var(X) = E(X2) – µ2 = 3.70 – (1.7)2 = 3.70 – 2.89 = 0.81. It follows that  = 0.81 = 0.9.

TI-83 – Means and Standard Deviations Store the list of values of X in L1. Store the list of probabilities of X in L2. Select STAT > CALC > 1-Var Stats. Press ENTER. Enter L1, L2. The list of statistics includes the mean and standard deviation of X. Use x, not Sx, for the standard deviation.

TI-83 – Means and Standard Deviations Let L1 = {0, 1, 2, 3}. Let L2 = {0.1, 0.3, 0.4, 0.2}. Compute the statistics. Compute µ and  for the Indoor and Outdoor distributions in Let’s Do It! 7.23, p. 430.

Let’s Do It! Return once more to Let’s Do It! 7.23, p. 430. The standard deviation of Profit Outdoors is 23.9. Use the original formula to compute the standard deviation of Profit Indoors. Use the alternate formula to compute the standard deviation of Profit Indoors. Use the TI-83 to find the standard deviation of Profit Indoors.