Random Variables & Probability Distributions

“The probability of someone laughing at you is proportional to the stupidity of your actions.”

DefinitionsRandom Variable: A variable whose possible values are numerical outcomes of a random phenomenon

Discrete Random Variable: Has a countable number of outcomes; distribution described by a histogram

Continuous Random Variable: Takes all values in an INTERVAL of numbers; distribution described by a density curve (i.e., normal curve)

Definitions

Sample Space:Set of all possible outcomes

Probability Distribution: A chart that lists the values of the discrete random variable X and the probability that X occurs; all probabilities add up to 1; can graph as a histogram

Expected Value of a Discrete Random Variable

Expected Value = mean =

*this is the sum of the products of x and P(x) from the probability distribution

Example 1: State whether each of the following random variables is discrete or continuous:a) The number of defective tires on a car.  discreteb) The body temperature of a hospital patient.  continuousc) The number of pages in a book.  discreted) The lifetime of a light bulb.

continuous

Example 2: Let X denote the number of broken eggs in a randomly selected carton of one dozen “store brand” eggs at a certain market. Suppose that the probability distribution of X is as follows:

 

a) Interpret P(X = 1) =.20.  The probability that there is 1 broken egg is .20.

b) Find the probability that there are exactly 4 broken eggs. P(X = 4) = .01 c) Find the probability that there are at least 2 broken eggs.  P(X = 2 or X = 3 or X = 4) = .10 + .04 + .01 = .15d) Find the probability that there are less than 2 broken eggs.  P(X = 0 or X = 1) = .65 + .20 = .85e) How many eggs should we “expect” to be broken? E(X) = 0(.65) + 1(.20) + 2(.10) + 3(.04) + 4(.01) =

X 0 1 2 3 4

P(X) .65 .20 .10 .04

Example 3. Suppose we flip a coin 3 times. Let X = the number of heads. a) Write the sample space of flipping the coin.

HHHHHTHTHTHHHTTTHTTTHTTT

Example 3. b) Determine the probability distribution.

  

X = # heads 0 1 2 3 Total

P(X) 1/8 3/8 3/8 1/8

Example 3. c) Sketch a histogram of the probability distribution. X = # heads 0 1 2 3 Total

P(X) 1/8 3/8 3/8 1/8

Example 3. d) How many heads would we “expect”?   

E(X) =

X = # heads 0 1 2 3 Total

P(X) 1/8 3/8 3/8 1/8

X*p(x) 1.5

Example 4: Suppose we toss two dice and sum the results. a) Write out the sample space     b) Determine the probability distribution.  c) What is the probability you get a sum less than 5? d) What is the probability you get a sum of at least 10? e) What is the expected value of the sum of the two dice?