06-Random Variables complete

06-Random Variables complete

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Unformatted text preview: area with five different toys. Over a fixed time period various observations are made. One response measured is the number of toys the child plays with. Based on many results, the probability distribution on the next page was determined for the general discrete random variable X = number of toys played with by children (during a fixed time period). 43 Psychology Experiment continued X = # toys Probability 0 0.03 1 0.16 2 0.30 3 0.23 4 0.17 5 a. What is the missing probability P(X = 5)? b. Graph this discrete probability distribution function for X. .50 .45 .40 .35 .30 .25 .20 Probability .15 .10 .05 0.00 0 X 1 2 3 4 5 = number of toys played with c. What is the probability that a child will play with at least 3 toys? Remember at least 3 is 3 or more, so just add up the corresponding probabilities. The more formal way to express it is: P(X 3) = 0.23 + 0.17 + 0.11 = 0.51 d. Given the child has played with at least 3 toys, what is the probability that he/she will play with all 5 toys? Think it through logically … look only at the values for 3 or more, out of this 0.51, we need the part that is for all 5 … so we have 0.11/0.51. The more formal way to write this is: P(X = 5 | X 3) = 0.11/0.51 = 0.216 e. Finish the table below to provide the cumulative distribution function of X. X = # toys 0 1 2 3 4 Cumulative Probability 0.03+0.16 0.03+0.16+0.30 0.03 0.72 0.89 = 0.19 = 0.49 P(X k) Note: the last box will always be 1.00. 44 5 1.00 8.3 Expectations for Random Variables Just as we moved from summarizing a set of data with a graph to numerical summaries, we next consider computing the mean and the standard deviation of a random variable. The mean can be viewed as the expected value over the long run (in many repetitions of the random circumstance) and the standard deviation can be viewed is approximately the average distance of the possible values of X from its mean. Definition: The expected value of a random variable is the mean value of the variable X in the sample space, or population, of possible outcomes. Expected value, denoted by E(X), can also be interpreted as the mean value that would be obtained from an infinite number of observations on the random variable. Motivation for the expected value formula … Consider a population consisting of 100 families in a community. Suppose that 30 families have just 1 child, 50 families have 2 children, and 20 families have 3 children. What is the mean or average number of children per family for this population? Mean = (sum of all values)/100 2 1 1 = [1(30) + 2(50) + 3(20)]/100 2 1 2 = 1(30/100) + 2(50/100) + 3(20/100) 2 2 3 = 1(0.30) + 2(0.50) + 3(0.20) 2 etc. 3 = 1.9 children per family Population of 100 families Mean = Sum of (value x probability of that value) Definitions: Mean and standard deviation of a discrete random variable Suppose that X is a discrete random variable with possible values x1, x2, x3, … occurring with probabilities p1, p2, p3, …, then the expected value (or mean) of X is given by = E(X) = the variance of X is given by V(X) = = (x i x p i i...
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This document was uploaded on 02/25/2014 for the course STATS 250 at University of Michigan.

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