Exercise 21 Page 913

Let's begin by recalling the definition of an expected value of a random variable.

An expected value is the average value of a random variable that is expected after repeating an experiment or a simulation an infinite number of times.

Therefore, we can evaluate an expected value by multiplying each possible value of a random variable X by the probability of that value occurring and then summing these products. Let's consider an example of rolling a die.

If a die is fair we have 6 possible outcomes, each with the probability of occurrence 16. Let's calculate the expected value of rolling a die. Let X represent a random variable describing the result of rolling a die once.

X	Probability	Product
1	1/6	1* 1/6
2	1/6	2* 1/6
3	1/6	3* 1/6
4	1/6	4* 1/6
5	1/6	5* 1/6
6	1/6	6* 1/6

Now let's evaluate the sum of the products.

E(X)=1*1/6+2*1/6+3*1/6+4*1/6+5*1/6+6*1/6

▼

Simplify right-hand side

MoveLeftFacToNumOne

a* 1/b= a/b

E(X)=1/6+2/6+3/6+4/6+5/6+6/6

AddFrac

Add fractions

E(X)=1+2+3+4+5+6/6

AddTerms

Add terms

E(X)=21/6

CalcQuot

Calculate quotient

E(X)=3.5

The expected value of rolling a die once is 3.5, which is not equal to any possible outcome of this experiment. This is because an expected value is the average, which does not have to be equal to any of the values from the data set.