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Unformatted text preview: ) 2 pi and so the standard deviation of X is given by = (x i ) 2 pi The sums are taken over all possible values of the random variable X. 45 Try It! Psychology Experiment Recall the probability distribution for the discrete random variable X number of toys played with by children. X = # toys Probability 0 0.03 1
0.23 4 0.17 5
0.11 a. What is the expected number of toys played with? = E(X) = 0(0.03) + 1(0.16) + 2(0.30) + 3(0.23) + 4(0.17) + 5(0.11) = 0 + 0.16 + 0.60 + 0.69 + 0.68 + 0.55 = 2.7 toys Does this number make sense? Recall the probability histogram. Note: The expected value may not be a value that is ever expected on a single random outcome. Instead, it is the average over the long run. b. What is the standard deviation for the number of toys played with? (x i ) 2 pi (0 2.7) 2 0.03 (1 2.7) 2 0.16 (5 2.7) 2 0.11 1.72 1.3 c. Complete the interpretation of this standard deviation (in terms of an average distance): On average, the number of toys played with vary by about ___1.3 ___ from the mean number of toys played with of __2.7 __. Or The average distance between the number of toys played with and the mean of 2.7 toys is approximately 1.3 toys. 46 8.4 Binomial Random Variables An important class of discrete random variables is called the Binomial Random Variables. A binomial random variable is that it COUNTS the number of times a certain event occurs out of a particular number of observations or trials of a random experiment. Examples of Binomial Random Variables: The number of girls in six independent births. The number of tall men (over 6 feet) in a random sample of 30 men from a large male population. A binomial experiment is defined by the following conditions: 1. There are n “trials” where n is determined in advance and is not a random value. 2. There are two possible outcomes on each trial, called “success” (S) and “failure” (F). 3. The outcomes are independent from one trial to the next. 4. The probability of a “success” remains the same from one trial to the next, and this probability is denoted by p. The probability of a “failure” is 1 – p for every trial. A binomial random variable is defined as X = number of successes in the n trials of a binomial experiment. Think about it: What are the possible values for X? 0, 1, 2, …, n Does it make sense that X is a discrete random variable? YES. Table 8.4 (page 276) has good examples and provides some good points to keep in mind when trying to decide whether a random variable fits the binomial description (read them). Try It! Are the Conditions Right for Binomial? a. Observe the sex of the next 50 children born at a local hospital. X = number of girls n is fixed in advance at 50, indep is ok (no multiple births), 2 outcomes (G, B), and p = 0.50. b. A ten‐question quiz has five True‐False questions and five multiple‐choice questions, each with four possible choices. A student randomly picks an answer for every question. X = number of answers that are...
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This document was uploaded on 02/25/2014 for the course STATS 250 at University of Michigan.
- Summer '10