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UGA STAT 4210 - Chapter 6

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Chapter 6 Probability Distributions A probability distribution does just that The total probability for an experiment or for all outcomes of a random process is 1 A probability distribution distributes that 1 total probability among all possible outcomes of the process Definition a random variable r v is a numerical measurement of a random process It maps the outcome of an experiment onto real numbers Example Flip a coin The sample space S H T You would observe therefore whether you got a H or a T But there isn t a random variable involved because H and T aren t numeric One possible r v to assign to this process would be to say let X be the number of H observed X 0 if no heads 1 if one head Now all possible outcomes of the sample space have been mapped onto R Note only r v s have probability distributions Definition a discrete r v X takes on a set of separate values e g 0 1 2 etc Its probability distribution p x assigns a probability to each distinct value of X The probability distribution for a discrete random variable is a listing of each possible value paired with its probability p x 0 x p x 1 x Pr X x p x Examples of discrete r v s the number of cars running red lights the outcome of a roulette spin score of a baseball game The mean or expected value of a probability distribution is the long run average of outcomes For a discrete random variable you get the mean by multiplying each value x by its probability p x and summing the results Therefore the expected value is a weighted average The mean because it is based on the probability distribution for the whole population is a parameter This is not necessarily a realizable number For a discrete r v mean E X xp x If the variables are categorical by nature and not inherently discrete we have to create an r v like we did with the coin toss If the categorical variable only has two groups of interest we can assign one outcome a success and the other failure and then count the successes Definition a continuous random variable can take on all possible values in an interval e g time between 12 00p and 12 01p The probability distribution of continuous r v f x assigns probabilitys to all intervals of possible values 1 f x 0 x R 2 f x dx 1 b 3 Pr a X b f x dx a That s more than we really need to know but the crux of it is this probabilities can t be less than 0 can t be more than 1 and the probability over the entire range of outcomes is 1 For continuous r vs the probability has to be found over an interval of possible values because probability is the area under a curve The probability of a continuous r v is the area under a curve f x There are infinite possible outcomes for a continuous r v in any interval In fact if you have a continuous r v X Pr X x 0 because the area of a line 0 the area under a curve at any point 0 We can only talk about probability over an interval The Normal and Standard Normal Distribution The most common continuous probability distribution is the normal distribution It forms the basis of the Empirical Rule The normal distribution is symmetric the mean median The standard deviation is the measure of spread The normal distribution is defined by these two parameters If X is a normal random variable with mean and standard deviation we write the following X N which says X follows the normal distribution with mean and standard deviation Because there are an infinite possible combinations of and there are and infinite number of possible normal distributions To simplify matters we utilize the standard normal Z distribution x x the number of sample standard deviations an s observation x is away from the from the sample mean x We can do the same thing at the population level using the parameters which will get us z scores or standardized normal random variables If X is a normal random variable we can use a similar transformation and get a standard normal random variable so that the mean is 0 and the standard deviation is 1 Recall standardized values z Definition the standard normal distribution Z distribution is a normal distribution with mean 0 and standard deviation 1 By using a similar conversion to the standardization above any normal r v X can be converted to a standard normal r v Z If X N Z X is a standard normal r v and Z N 0 1 Example Husky weights follow a normal distribution and the average weight of female huskies is 42 lbs with a standard deviation of 0 67lb Sikari weights 40 lbs What is her z score z x 40 42 2 2 985 67 67 Interpret this value Sikari weighs 2 986 standard deviations less than the average female husky What do we do whit this information We could use the Empirical Rule but that only gets us so far because it only tells us about probabilities within 1 2 and 3 standard deviations of the population mean What if I want to know about specific intervals Use the standard normal table As the plot at the top of the page Appendix A shows the standard normal table provides cumulative probabilities for z scores It is the probability of obtaining any observation less than a given z score They are all left tail probabilities It works in two ways 1 Given a z score find the cumulative probability percentile 2 Given a cumulative probability find the associated z score Example What is the probability of finding a female husky that weighs less than Sikari Pr X 40 Pr X 40 42 Pr Z 2 985 0 0014 67 Example What is the probability of finding a female husky that weighs more than Sikari Pr X 40 1 Pr X 40 1 0 0014 0 9986 Example What is the probability of finding a female husky with a weight closer to the average than Sikari s 40 42 X 44 42 Pr 2 985 Z 2 985 67 67 Pr Z 2 985 Pr Z 2 985 9986 0014 9972 Pr 40 X 44 Pr Example There is a female husky that weighs more than 40 9 of other female huskies How much does she weigh Pr Z z 0 409 Pr Z 0 23 0 409 Pr Z 23 0 409 Pr X 41 846 0 409 x 41 846 lbs When you have categorical data that can be classified into two groups male female blonde not blonde and you have independent trials you can use the binomial distribution to understand the probability of observing the number of successes in n trials We say those trials are binary because there are two outcomes Conditions for Binomial Distribution each of n trials …


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UGA STAT 4210 - Chapter 6

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