MATH 2565 - Lecture 10

Random Variables (Chapter 4.3 & 4.4)

What is a random variable?

A variable that takes on numerical values according to outcomes of a random phenomenon

What is a discrete random variable?

A variable that takes on a countable number of possible values

What is a continuous random variable? (Ex: (Height - (170.2 cm, 170.23 cm, etc.))

A variable that takes on all values in an interval

The __________ ____________ of a discrete random variable is the collection of probabilities for all possible values of the random variable

Probability distribution

The ________ ______ of a random variable is the theoretical (population) mean of the random variable

Expected value

Calculate the expected value of Y = the number of heads obtained after tossing a coin three times

1.5

What is the probability that at least 2 heads are observed?

0.5

Calculate the variance and the standard deviation of Y = the number of heads obtained after tossing a coin three times

Calculate the expected value and the standard deviation of X.

Expected Value = 2.5

Standard Deviation = 1.204

Find the expected value and the variance of X

Expected Value = 1.2

Variance = 0.48

What is your average payoff from playing many tickets and its interpretation? (hint: µ_x)

$0.50 - In the long run, you expect to win an average of 50 cents per ticket

Draw independent observations at random from any population with finite mean µ. As the number of observations n increases, the sample mean ̅x gets closer and closer to the theoretical mean µ.

This statement describes…

The Law of Large Numbers

Calculate the expected value and variance for the total annual reimbursement amount per client for massage and acupuncture treatments.

Expected Value = 168

Variance = 9720