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Unformatted text preview: 6 Probability Distributions 6.1 Summarizing Possible Outcomes and Their Probabilities Random Variable A random variable is a numerical measurement of the outcome of a random phenomenon. Often, the randomness results from the use of random sampling or a randomized experiment to gather the data. Ex.1: Number of touchdowns for Red Raiders, Duration of a call Probability Distribution The probability distribution of a random variable specifies its possible values and their probabilities. Ex.2: Probability distribution of number of home runs in a game for Boston Red Sox. The table lists the possible values for the number of home runs and the corresponding probabilities. Number of Home Runs Probability 0.23 1 0.38 2 0.22 3 0.13 4 0.03 5 0.01 6 or more 0.00 Probability distributions can be defined for both discrete random variables and continuous random variables. We will first look at discrete random variables. Probability Distribution of a Discrete Random Variable A discrete random variable X takes a set of seperate values (such as 0,1,2,....). Its probability distribution assigns a probability P ( x ) to each possible value of x . • For each x , the probability P ( x ) falls between 0 and 1. • The sum of the probabilities for all the possible x values equals 1. 1 Ex.3: Refer to Ex.2 . • for each x , the probability falls between 0 and 1. (all the values 0.23, 0.38, 0.22, 0.13, 0.03, and 0.01 are between 0 and 1) • sum of the probabilities for all the possible x values equals 1. (0.23 + 0.38 + 0.22 + 0.13 + 0.03 + 0.01 = 1) Therefore, it is a probability distribution. Probability of at least 3 homeruns is P ( X ≥ 3) = P (3) + P (4) + P (5) + P (6) = 0 . 13 + 0 . 03 + 0 . 01 + 0 . 00 = 0 . 17 Figure 1: Probability Distribution for Ex.2 Ex.4: From six marbles numbered as 1,1,1,1,2,2, two marbles will be drawn at ran dom without replacement. Let X denote the sume of the numbers on the selected marbles. List the possible values of X and determine the probability distribution. 2 X sum of the two numbers The probability distribution for this experiment is given in the following table. X Probability 2 6 15 3 8 15 4 1 15 The Mean of a Probability Distribution ( μ ) The mean of a probability distribution for a discrete random variable is μ = summationdisplay xP ( x ) where the sum is taken over all possible values of x . This is also called the expected value for X . Ex.5: The mean of the number of homeruns for in a game for Red Sox is, μ = 0(0 . 23) + 1(0 . 38) + 2(0 . 22) + 3(0 . 13) + 4(0 . 03) + 5(0 . 01) = 1 . 38 This also means that you can expect the Red Sox to have an average of 1.38 homeruns per game for that season. Probability Distribution of a Continuous Random Variable A continuous random variable has possible values that form an interval. Its probability distribution is specified by a curve that determines the probability that the random variable falls in any particular interval of values....
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 Spring '08
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 Statistics, Normal Distribution, Probability, Probability theory

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