1. 4.1 Probability Distributions Important Concepts –Random Variables –Probability Distribution –Mean (or Expected Value) of a Random Variable –Variance and Standard Deviation of a Random Variable

2. 4.1 Probability Distributions Consider the following experiment: –Suppose we toss a coin three times. What is the sample space for this experiment? What is the probability of tossing exactly: 0 tails? 1 tails? 2 tails? 3 tails?

3. 4.1 Probability Distributions Terms to know: –A random variable X represents a numerical value associated with each outcome of a probability experiment. A random variable is discrete if it has a finite or countable number of possible outcomes. A random variable is continuous if it has an uncountable number of possible outcomes.

4. 4.1 Probability Distributions Terms to know: –A–A discrete probability distribution lists each possible value a random variable can assume, together with its probability. All probability distributions must satisfy the following two conditions: –T–The probability of each value of the random variable must be between 0 and 1, inclusive. –T–The sum of all the probabilities must be 1. #26 p. 198 #28 p. 198

5. # of tails, xP( X = x ) 01/8 = 0.125 13/8 = 0.375 2 31/8 = 0.125 1.000 4.1 Probability Distributions Discrete probability distribution of our random variable X:

6. 4.1 Probability Distributions How do we find the mean and standard deviation of a discrete random variable? In chapter 3, we used the following: Can we still use these formulas?

7. Let’s try #31 p. 199 (Camping Chairs) Number of Cats, x P(X=x)x·P(X=x)x2x2 x 2 ·P(X=x) 00.250000 10.298 1 20.2290.45840.916 30.1680.50491.512 40.0340.136160.544 50.0210.105250.525 1.5013.795

8. 4.1 Probability Distributions #33 p. 199 (Hurricanes) Category, xP(X = x)x∙P(X =x)x2x2 x 2 ∙P(X =x) 10.418 1 20.2610.52241.044 30.2470.74192.223 40.0630.252161.008 50.0100.05250.25 1.9834.943