Probability*Distributions

1 3 probability Distributions(Ch , , , , )2 probability Distribution FunctionsProbability distribution function(pdf): Function for mapping random variables to real numbers. Discrete random variable: Values constitute a finite or countablyinfinite random variable:Set of possible values is the set of real numbers R, one interval, or a disjoint union of intervals on the real line. 3 Random VariablesNotation! variables - usually denoted by uppercase letters near the end of our alphabet ( X, Y).

2 Value - now use lowercase letters, such as x, which correspond to the !"=$=1&''()( + ,$-$=1&''()( + Properties of PDFsFor f(x) to be a legitimate pdf, it must satisfy the following two (x) 0 for all discrete continuous Random Variables6 The pdf of a discrete X describeshow the total probability is distributed among all the possible range values of the X:f(x) = p(X=x), for each value x in the range of XPDFs for Discrete RVs7 Example A lab has 6 computers.))

3 Let X denote the number of these computers that are in use during lunch hour - - {0, 1, 6}. Suppose that the probability distribution of X is as given in the following table:8 Example From here, we can find many that at most 2 computers are in that at least half of the computers are in that there are 3 or 4 computers free9 Bernoulli DistributionBernoulli random variable: Any random variable whose only possible values are 0 or is a discrete random variable why?

4 This distribution is specified with a single parameter:P(X = x) = x(1- )(1- x); x = 0, 110 Bernoulli DistributionBernoulli random variable: Any random variable whose only possible values are 0 or is a discrete random variable why?This distribution is specified with a single parameter:P(X = x) = x(1- )(1- x); x = 0, 1 Examples?11 Geometric DistributionA patient is waiting for a suitable matching kidney donor for a transplant. The probability that a randomly selected donor is a suitable match is What is the probability the first donor tested is the first matching donor?

5 Second? Third?12 Continuing in this way, a general formula for the pmfemerges:The parameter can assume any value between 0 and 1. Depending on what parameter is, we get different members of the geometricdistribution. NOTATION: We write X ~ G( ) to indicate that X is a geometric rvwith success probability . Geometric Distribution!"=$=(1 ()*(;,,,,,,$=0,1,2,.., 13 The Binomial counts the total number of successesout of n trials, where Xis the number of successes. Each trial must be independent of the previous experiment.))

6 The probability of success must be the same for each : A dice is tossed four times. A success is defined as rolling a 1 or a 6. The probability of success is 1/3. What is P(X = 2)? What is P(X = 3)?Binomial Distribution15 Example: A dice is tossed four times. A success is defined as rolling a 1 or a 6. The probability of success is 1/3. What is P(X = 2)? What is P(X = 3)?Let s use the probabilities we calculated above to derive the binomial Distribution16 Example: A dice is tossed four times.

7 A success is defined as rolling a 1 or a 6. The probability of success is 1/3. What is P(X = 2)? What is P(X = 3)?Let s use the probabilities we calculated above to derive the binomial : We write X ~ Bin(n, ) to indicate that X is a binomial rvbased on n Bernoulli trials with success probability . Binomial Distribution 17 The Negative Binomial DistributionConsider the dice example for the binomial distribution. Now we instead want to find the probability that we roll 3 failures ( a 2, 3, 4, or 5) before the is this related to the binomial distribution?

8 18 The Negative Binomial DistributionConsider the dice example for the binomial distribution. What is the probability that exactly 3 successes occur before 2 failures occur?NOTATION: We write X ~ NB(r, ) to indicate that X is a negative binomial , with xfailures occurring before rsuccesses, where the probability of success is equal to . 19 The Poisson probability DistributionA Poisson describes the total number of events that happen in a certain time : - # of vehicles arriving at a parking lot in one week- # of gamma rays hitting a satellite per hour- # of cookies chips in a length of cookie dough20 The Poisson probability DistributionA Poisson describes the total number of events that happen in a certain time.

9 - # of vehicles arriving at a parking lot in one week- # of gamma rays hitting a satellite per hour- # of cookies sold at a bake sale in 1 hour21 The Poisson probability DistributionA Poisson describes the total number of events that happen in a certain time discrete random variable X is said to have a Poisson distribution with parameter ( > 0) if the pdf of X isNOTATION: We write X ~ P( ) to indicate that X is a Poisson with parameter .!"=$=% %%%%%%%$=0,1,2.

10 22 Example Let X denote the number of mosquitoes captured in a trap during a given time that X has a Poisson distribution with = What is the probability that the trap contains 5 mosquitoes? 23 Example problem24 Cumulative Distribution FunctionsDefinition: The cumulative distribution function (cdf) is denoted with F(x).For a discrete with pdf f(x) = P(X = x), F(x) is defined for every real number x byFor any number x, the cdf F(x) is the probabilitythat the observed value of X will be at mostx.