Statistics 312 10 Probability Distributions 1 A probability distribution for a discrete random variable is a linking of all its possible outcomes and their probabilities Since all possible outcomes are listed the sum of the probabilities must add to 1 0 Example Coin flips Suppose we let the random variable be X the number of heads in three flips of a fair coin Then P HHH 1 8 P HHT 1 8 P HTH 1 8 P THH 1 8 P TTH 1 8 P THT 1 8 P HTT 1 8 P TTT 1 8 x 0 1 2 3 p x 1 8 3 8 3 8 1 8 2 6 27 3 1 27 Suppose coin is weighted with P HHH 1 27 P HHT 2 27 P HTH 2 27 P THH 2 27 P TTH 4 27 P THT 4 27 P HTT 4 27 P TTT 8 27 x p x 0 8 27 1 12 27 Both satisfy the definition of a probability distribution because all outcomes 0 1 2 and 3 are listed and the sum of the probabilities equals 1 0 The Expected Value or Average of a Random Variable The mean x of a probability distribution is called the expected value of the random variable N x E X X i P X i i 1 Statistics 312 10 Probability Distributions 2 Example Coin Flips x p x 0 1 8 1 3 8 2 3 8 3 1 8 x 0 1 8 1 3 8 2 3 8 3 1 8 12 8 3 2 1 5 Variance and Standard Deviation of a Random Variable The variance of a N random variable is 2x X i x 2 P X i i 1 In addition the standard deviation x of the probability distribution of a random variable is the square root of the variance and is given by Example Coin Flips x p x 0 1 8 1 3 8 2 3 8 x 0 3 2 1 8 1 3 2 3 8 2 3 2 3 8 3 3 2 1 8 24 32 3 4 75 x 75 866 Read pp 144 148 Prob 4 8 4 9 4 11 4 12 3 1 8
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