CK-12 Probability and Statistics - Advanced

4.3. Mean and Standard Deviation of Discrete Random Variables http://www.ck12.org

μ= ( 0 )( 99 .9%)+( 15 , 000 )( 0 .1%)

= 0 + 15

μ=$15

Since the company charges $310 and expects to pay out $15, the profit for the company is $295 on every policy.

Sometimes, we are interested in measuring not just the expected value of a random variable but also thevariability
and thecentral tendencyof a probability distribution. To do so, we first define thepopulation variance,σ^2. It is
defined as the average of the squared distance of the values of the random variablexto the mean valueμ. The formal
definitions of the variance and the standard deviation are shown below.

The Variance

The variance of a discrete random variable is given by the formula

σ^2 =∑

x

(x−μ)^2 p(x)

The Standard Deviation

The square root of the varianceσ^2 is the standard deviation of a discrete random variable,

σ=

√

σ^2

Example:

A university medical research center finds out that treatment of skin cancer by the use of chemotherapy has a success
rate of 70%. Suppose five patients are treated with chemotherapy. If the probability distribution ofxsuccessful cures
of the five patients is given in the table below:

x 0 1 2 3 4 5

p(x) 0. 002 0. 029 0. 132 0. 309 0. 360 0. 168

Figure:Probability distribution of cancer cures of five patients.

a) Findμ

b) Findσ

c) Graphp(x)and explain howμandσcan be used to describep(x).

Solution:

a. We use the formula