# Probability Distributions Random Variables Discrete Probability Distributions Mean

- Slides: 11

Probability Distributions Random Variables * Discrete Probability Distributions * Mean, Variance, and Standard Deviation * Expected Value

Fill in the blanks 1 – 3 1. Random Variable: A count or an ________ outcome of a probability experiment. (Uses numerical values) 2. Discrete random variable: Has a finite or countable _________ number of possible outcomes that can be listed. noncountable 3. Continuous random variable: Has a _______ number of possible outcomes, represented by an interval on the number line.

Problem # 4 possible 4. Discrete probability distribution: Lists each _______ value that the random variable can assume, together with its probability. The following conditions must apply: a. The probability of each value of the discrete random variable is between 0 and 1, inclusive. {0, 1} i. ____________ b. The sum of all the probabilities is 1. 1 i. = _________

5 - 8 5. Mean of a discrete random variable: Each multiplied probability of x is _____ by its corresponding probability and the products are added. *Mean values may be bigger than 1 since they are for more than one person! 6. Variance of a discrete random variable is 7. Standard Deviation is 8. Expected Value: A ______ random variable is equal discrete to the mean of the random variable. Expected value = (note that the mean and expected values are the same!)

Example #1 Determine the missing probability value x P(x) 0 0. 05 1 0. 23 2 0. 10 3 4 ? 0. 35 0. 11 5 0. 16

Example #2 Constructing a Discrete Probability Distribution 1. Make a frequency distribution for the possible outcomes. 2. Find the sum of the frequencies. 3. Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies. 4. Check that each probability is between 0 and 1 and that the sum is 1. Score, x Frequency, f Relative frequency P(x) 1 30 0. 21 2 31 0. 22 3 12 0. 09 4 28 0. 20 5 39 0. 28 140 1

Example #3 NO! it adds Example #3 Is the following a probability distribution? _______ up to 1. 1 x 0 1 2 3 4 P(x) 0. 40 0. 10 0. 25 0. 30 . 05

Example #4 Dogs 0 Households 1491 1 2 3 4 5 425 168 48 29 14 Hint: Use the lists in your calculator!

Example #4 Continued

More Example #4 0. 502 b) Find the mean ____________ c) Find the variance __________ 0. 826 0. 909 d) Find the standard deviation ___________ e) A household on average has _____ dogs with a 0. 5* standard deviation of ________. 0. 9 Note, if it is a nice dog I want the front half!

Example #5 A charity organization is selling $4 raffle tickets as part of a fund-raising program. The first prize is a boat valued at $3150, and second prize is a camping tent valued at $450. The rest of the prizes are 15 - $25 gift certificates. The number of tickets sold is 5000. Find the expected net gain to the player for one play of the game. Is the players expected to win or to lose?

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