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# Probability Distributions

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1. Probability Distributions ROCKS!!!!!!!!

2. Agenda • Probability Distributions • Binomial Distributions • Geometric Distributions • Hypergeometric Distributions

3. Probability Distribution • The probability distribution consists of a table of values of X=x and P(x) or a graph. • Individual trials have specific probabilities.

4. Probability Distributions • The random variable X has a single value for each outcome within an experiment • Discrete variables have separate and equi-distant values from each other •  Ex. 1, 2, 3, and 4 • Continuous variables have an infinite number of possible values  Ex. 1.2367 • The probability that X will take on a particular value is P(x) • If all the values are equally likely than you have a discrete uniform distribution • The formula is P(x) = 1/n • An expected value is the average of all possible outcomes of a probability experiment  E(x) = x1 P(x1) + x2 P(x2) … + xn P(xn)

5. Binomial Distributions • Has a specified number of independent trials in which the outcome is either success or failure • The Random Variable is the number of successes in a given number of trials. • Probability of x successes in n independent trials is: P(x)= nCxp^xq^n-x • The expectation for a binomial distribution is E(x)=np

6. Geometric Distributions • Has a specified number of independent trials with two possible outcomes, success or failure. • Random variable is the number of unsuccessful outcomes before a success occurs. • Probability of success after a waiting time of x failures is P(x)= q^x (p) • Expectation of a geometric distribution is E(X)= q/p • Must ensure that the probability on a single trial is accurate for the situation and that each trial is independent • Summarize the results by calculation probabilities and the expected waiting time.

7. Hypergeometric Distributions • Has a specified number of dependent trials having two possible outcomes, success or failure • Random variable is number of successful outcomes cannot be repeated within these trials • Probability of x successes in r dependent trials is P(x)= aCx Xn-a Cr-x/ nCr • The expectation for this is E(X)= ra/n • Ensure that number of trials is representative of this situation and that each trial is dependent • Record number of successes and summarize the results by calculating probabilities and expectation