Chapter 3 - Lecture 3 Expected Values of Discrete Random Variables October 5th, 2009
Properties of expected value Standard deviation Shortcut formula Properties of the variance
Properties of expected value In State College there are 10000 families. I asked them how many kids they have and I get the following answers: 0 kids: 900 families 1 kid: 1700 families 2 kids: 4000 families 3 kids: 2100 families 4 kids: 700 families 5 kids: 400 families 6 kids: 200 families If I choose a family at random, how many kids I expect it to have? How did I calculate this?
Properties of expected value Expected value If X is a random variable with a set of all possible values denoted with D and pmf p(x). The expected value or mean value of X is: E(X ) = µ X = x D xp(x)
Properties of expected value Roll of a die I roll a die once. What is the expected value of the outcome?
Properties of expected value Birth of a child Let say the probability of a pregnant woman to give a birth to a boy is p. What is the expected number of X = the child is a boy? What is the expected number of Y = the child is a girl? How many boys I expect to see if I observe 100 births?
Properties of expected value If the random variable X has a set of possible values D then the expected value of the function of X, h(x ), is calculated by: E(h(X )) = µ h (X ) = x D h(x)p(x)
Properties of expected value Lets go back to the example with the kids per family in State College. Let s say that each family spends Y = X 2 + 5 X + 100 dollars on grocery shopping each week where X is the number of kids in the family. How much money on average families in State College spend in grocery shopping per week.
Properties of expected value Properties E(aX ) = ae(x ) Proof? E(X + b) = E(X ) + b Proof? E(aX + b) =?
Outline Standard deviation Shortcut formula Properties of the variance Assume that I give to 5 students a test and their scores are 85, 85, 85, 85, 85. What is the average of those students? Assume that I give to 5 students a test and their scores are 50, 85, 100, 100, 90. What is the average of those students?
of a random variable Standard deviation Shortcut formula Properties of the variance Let X have pmf p(x) and expected value E(X ). Then the variance of X is given by: ( var(x ) = σx 2 = E (X E(X )) 2) = (x E(X )) 2 p(x) x D
Standard deviation Outline Standard deviation Shortcut formula Properties of the variance The standard deviation is another useful measure and is actually the square root of the variance: σ X = σx 2
Outline Standard deviation Shortcut formula Properties of the variance Lets go back to the example with the kids per family in State College. Find the variance and standard deviation of the number of kids per family in State College.
Shortcut formula for Standard deviation Shortcut formula Properties of the variance
Properties Outline Standard deviation Shortcut formula Properties of the variance var(ax ) = a 2 var(x ) Proof? var(x + b) = var(x ) Proof? var(ax + b) =?
Section 3.3 page 116 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43