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Expected Value. Section 7.4 (partially). Section Summary. Expected Value Linearity of Expectations Independent Random Variables. Expected Value. Definition : The expected value (or expectation or mean ) of the random variable X ( s ) on the sample space S is equal to

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expected value

Expected Value

Section 7.4 (partially)

section summary
Section Summary
  • Expected Value
  • Linearity of Expectations
  • Independent Random Variables
expected value1
Expected Value

Definition: The expected value (or expectation or mean) of the random variable X(s) on the sample space S is equal to

Example-Expected Value of a Die: Let X be the number that comes up when a fair die is rolled. What is the expected value of X?

Solution: The random variable X takes the values 1, 2, 3, 4, 5, or 6. Each has probability 1/6. It follows that

expected value2
Expected Value

Theorem 1: If X is a random variable and p(X = r) is the probability that X = r, so that

then

see the text for the proof

Example: what is the expected sum of the numbers that appear

when two fair dice are rolled?

Hint: compute p(X=k) for each k from 2 to 12 and use Theorem 1

to get expected sum 7.

expected value3
Expected Value

Theorem 2: The expected number of successes when n mutually independent Bernoulli trials are performed is np, where p is the probability of success on each trial.

see the text for the proof

Example: what is the expected number of heads that come up when a

fair coin is flipped 5 times.

Solution: By Theorem 2 with p=1/2 and n=5, we see that the expected

number of heads is 2.5 .

linearity of expectations
Linearity of Expectations

The following theorem tells us that expected values are linear. For example, the expected value of the sum of random variables is the sum of their expected values.

Theorem 3: If Xi, i = 1, 2, …,n with n a positive integer, are random variables on S, and if a and b are real numbers, then

(i) E(X1 + X2 + …. + Xn) = E(X1 )+ E(X2) + …. + E(Xn)

(ii) E(aX + b) = aE(X) + b.

see the text for the proof

independent random variables
Independent Random Variables

Definition 3: The random variables X and Y on a sample space S are independent if

p(X = r1 and Y = r2) = p(X = r1)∙p(Y = r2).

Theorem 5: If X and Y are independent variables on a sample space S, then E(XY) = E(X)E(Y).

see text for the proof

independent random variables1
Independent Random Variables

Theorem 5: If X and Y are independent variables on a sample space S, then E(XY) = E(X)E(Y).

Example: Let X be the number that comes up on the first die when two fair dice are rolled and Y be the sum of the numbers appearing on the two dice. Show that E(X)E(Y) is not equal E(XY).

Answer: E(X)=7/2. E(Y)= 7. E(XY)= 329/12.

For example,