Note: A probability distribution with a small (large) spread about its mean will have a small (large) variance.
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1 Section 8.3: Variance and Standard Deviation The Variance of a random variable X is the measure of degree of dispersion, or spread, of a probability distribution about its mean (i.e. how much on average each of the values of X deviates from the mean). Note: A probability distribution with a small (large) spread about its mean will have a small (large) variance. Variance of a Random Variable X Suppose a random variable has the probability distribution ( = ) and expected value ( ) =. Then the variance of the random variable X is ( )= ( ) + ( ) + + ( ) Note: We square each since some may be negative. Standard Deviation measures the same thing as the variance. The standard deviation of a Random Variable X is = Var( )= ( ) + ( ) + + ( ) Example 1: Compute the mean, variance and standard deviation of the random variable X with probability distribution as follows: X P(X=x)
2 Example 2: An investor is considering two business ventures. The anticipated returns (in thousands of dollars) of each venture are described by the following probability distributions: Venture 1 Venture 2 Earnings Probability Earning Probability a. Compute the mean and variance for each venture. b. Which investment would provide the investor with the higher expected return (the greater mean)? c. Which investment would the element of risk be less (that is, which probability distribution has the smaller variance)? Popper 1: The probability distribution of a random variable X is given below. Given the mean = 5.2 X P(X=x) 1/3 1/10 2/15 7/30 1/5 Find the variance. a b. 42 c d
3 Chebychev s Inequality Let X be a random variable with expected value and standard deviation. Then, the probability that a randomly chosen outcome of the experiment lies between and + is at least 1 ; that is, ( + ) 1 Let s see what this is saying: Example 3: A probability distribution has a mean 20 and a standard deviation of 3. Use Chebychev s Inequality to estimate the probability that an outcome of the experiment lies between 12 and 28. 3
4 Example 4: A light bulb has an expected life of 200 hours and a standard deviation of 2 hours. Use Chebychev s Inequality to estimate the probability that one of these light bulbs will last between 190 and 210 hours? Popper 2: A light bulb at an art museum has an expected life of 300 hours and a standard deviation of 12 hours. Use Chebychev's Inequality to estimate the probability that one of these light bulbs will last between 280 and 320 hours of use. a b c d
5 Math 1313 Section 8.4 Section 8.4: The Binomial Distribution A binomial experiment has the following properties: 1. Number of trials is fixed. 2. There are 2 outcomes of the experiment. Success, probability denoted by, and failure, probability denoted by. Note + =1 3. The probability of success in each trial is the same. 4. The trials are independent of each other. Experiments with two outcomes are called Bernoulli trials or Binomial trials. Finding the Probability of an Event of a Binomial Experiment: In a binomial experiment in which the probability of success in any trial is p, the probability of exactly x successes in n independent trials is given by = =, X is called a binomial random variable and its probability distribution is called a binomial probability distribution. Example 1 on page 505 derives this formula. Example 1: Consider the following binomial experiment. A fair die is cast four times. Compute the probability of obtaining exactly one 6 in the four throws. 1
6 Math 1313 Section 8.4 Example 2: Let the random variable X denote the number of girls in a five-child family. If the probability of a female birth is 0.5, construct the binomial distribution associated with this experiment. What is the probability of having at least one girl? Example 3: Consider the following binomial experiment. If the probability that a marriage will end in divorce within 20 years after its start is 0.6, what is the probability that out of 6 couples just married, in the next 20 years all will be divorced? a. None will be divorced? b. Exactly two couples will be divorced? c. At least two couples will be divorced? 2
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