Lesson 6 Probability Distributions 1 Distributions 2 2 Binomial Random Variables 6 3 Normal Random Variables 8 www apsu edu jonesmatt 1530 htm 1 1 Distributions A random variable is a quantitative variable whose value depends on chance We ll use them to model the behavior of populations A discrete random variable is a random variable whose possible values form a finite or countably infinite set of numbers A continuous random variable is a random variable whose possible values form a continuum or interval of numbers Use capital letters for random variables and lower case letters for their possible values The distribution of a random variable is TWO things 1 a list of the possible values it can take on 2 a list of the probabilities of those possible values www apsu edu jonesmatt 1530 htm 2 Example 1 Let X of siblings for a person chosen at random of Siblings x 0 1 2 3 4 P X x 0 2 0 425 0 275 0 075 0 025 X is the random variable representing the number of siblings x represents a possible value for the of siblings and can be 0 1 2 3 or 4 P X x is the probability that the number of children equals x For any discrete random variable X X P X x 1 x because the events X x are disjoint and the probability of getting some outcome in the sample space is 1 www apsu edu jonesmatt 1530 htm 3 Mean of a Random Variable The mean or expected value of X is X E X xP X x x The mean is a weighted average of the possible values of x where the weights are the probabilities If the distribution of X describes a population the sample means of large samples tend to be close to the mean Example 2 Find the expected number of siblings for a person chosen at random of Siblings x 0 1 2 3 4 P X x 0 2 0 425 0 275 0 075 0 025 www apsu edu jonesmatt 1530 htm 4 Standard Deviation of a Discrete Random Variable If there are N equally likely possible outcomes for a random variable X so each occurs with probability 1 N the standard deviation of X is given by rP s 2 X x x x 2 1 N N x If the outcomes are not equally likely this generalizes to s X x 2 P X x x Example 3 Find the standard deviation of the number of siblings of Siblings x 0 1 2 3 4 P X x 0 2 0 425 0 275 0 075 0 025 www apsu edu jonesmatt 1530 htm 5 2 Binomial Random Variables X is a binomial random variable with parameters n and p if n x P X x p 1 p n x x where x is any number in 0 1 2 n X denotes the number of successes that occur in n independent trials where each trial has success probability p Example 4 There are 49 senators each of whom shows for the senate meeting with probability 0 95 What is the probability that no one shows What is the probability that everyone shows What is the probability that exactly 46 show What is the probability that at least 47 show www apsu edu jonesmatt 1530 htm 6 Mean and Standard Deviation of a Binomial Random Variable If X is a binomial random variable with parameters n and p its mean is given by np and its standard deviation is given by p np 1 p Example 5 Of the 49 senators each showing up with probability 0 95 independently of the others what is the mean number who show What is the standard deviation of the number who show www apsu edu jonesmatt 1530 htm 7 3 Normal Random Variables A random variable has the normal distribution with mean and standard deviation if its density function is x 2 1 f x e 2 2 2 where is the mean and is the standard deviation The graph of the density function looks like this www apsu edu jonesmatt 1530 htm 8 More about Normal Random Variables Normal random variables arise in the study of natural and industrial systems Many statistical tests are based on normal distributions If X has a normal distribution with mean and standard deviation then P a X b is the area of the region under the normal curve between the points a and b above the x axis If X is distributed normal with mean and standard deviation use the calculator to find probabilities by doing P a X b normalcdf a b www apsu edu jonesmatt 1530 htm 9 Example 6 The amount of coffee X your statistics professor drinks each day has a normal distribution with mean 55 oz and standard deviation 6 oz Shade the appropriate regions beneath the density function and find the following P 49 X 55 P 55 X 61 P X 55 P X 65 P X 45 or X 65 P X 55 12 www apsu edu jonesmatt 1530 htm 10 Empirical Rule The empirical rule states if X is distributed normal with mean and standard deviation then P X 68 27 P X 2 95 45 P X 3 99 73 Another way to think about this for a normally distributed population with mean and standard deviation approximately 68 27 of all observations are within one standard deviation of the mean 95 45 of all observations are within two standard deviations of the mean 99 73 of all observations are within three standard deviations of the mean www apsu edu jonesmatt 1530 htm 11 Example 7 The heights of men in the United States are distributed approximately normal with mean 69 inches and standard deviation 2 8 inches Then the empirical rule tells us Example 8 Ball bearings have masses distributed normal with mean 1 2 grams and standard deviation 0 03 grams Then the empirical rule tells us www apsu edu jonesmatt 1530 htm 12 Standard Normal Random Variables A standard normal random variable Z has a normal distribution with mean 0 and standard deviation 1 Setting 0 and 1 in the density function on Slide 2 gives the density function for a standard normal random variable 1 z2 2 f z e 2 The graph of the density function looks like this If Z is distributed normal with mean 0 and standard deviation 1 use your calculator to find probabilities as P a Z b normalcdf a b 0 1 www apsu edu jonesmatt 1530 htm or simply normalcdf a b 13 Relationship Between Standard Normals and Other Normals If X is distributed normal with mean and standard deviation then X has a normal distribution with mean 0 and standard deviation 1 That is for any numbers a and b b a Z P a X b P This observation allows us to use normal tables to obtain probabilities for any normal random variable www apsu edu jonesmatt 1530 htm 14 Example 9 Suppose X normal 5 2 Calculate the following with your calculator …