# Random Variables & Probability Distributions The probability of someone laughing at you is proportional to the stupidity of your actions.

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Random Variables & Probability Distributions The probability of someone laughing at you is proportional to the stupidity of your actions.

Definitions Random 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

Example 1: State whether each of the following random variables is discrete or continuous: a) The number of defective tires on a car. discrete b) The body temperature of a hospital patient. continuous c) The number of pages in a book. discrete d) 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 =.15 d) Find the probability that there are less than 2 broken eggs. P(X = 0 or X = 1) =.65 +.20 =.85 e) How many eggs should we expect to be broken? E(X) = 0(.65) + 1(.20) + 2(.10) + 3(.04) + 4(.01) = X01234 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. HHH HHT HTH THH HTT THT TTH TTT

Example 3. b) Determine the probability distribution. X = # heads0123Total P(X)1/83/8 1/8

Example 3. c) Sketch a histogram of the probability distribution. X = # heads0123Total P(X)1/83/8 1/8

X = # heads0123Total P(X)1/83/8 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?

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