University of California, Los AngelesDepartment of StatisticsStatistics 100A Instructor: Nicolas ChristouSome special discrete probability distributions• Bernoulli random variable:It is a variable that has 2 possible outcomes: “success”, or “fail-ure”. Success occurs with probability p and failure with proba-bility 1 − p.1• Binomial probability distribution:Suppose that n independent Bernoulli trials each one havingprobability of success p are to be performed. Let X be thenumber of successes among the n trials. We say that X followsthe binomial probability distribution with parameters n, p.Probability mass function of X:P (X = x) =nxpx(1 − p)n−x, x = 0, 1, 2, 3, · · · , norP (X = x) = nCx px(1 − p)n−x, x = 0, 1, 2, 3, · · · , nwherenCx =nx=n!(n − x)!x!Expected value of X: E(X) = npVariance of X: σ2= np(1 − p)Standard deviation of X: σ =rnp(1 − p2• Geometric probability distribution:Suppose that repeated independent Bernoulli trials each one hav-ing probability of success p are to be performed. Let X be thenumber of trials needed until the first success occurs. We saythat X follows the geometric probability distribution with pa-rameter p.Probability mass function of X:P (X = x) = (1 − p)x−1p, x = 1, 2, 3, · · ·Expected value of X: E(X) =1pVariance of X: σ2=1−pp2Standard deviation of X: σ =s1−pp23• More on geometric probability distribution · · ·Repeated Bernoulli trials are performed until the first successoccurs. Find the probability that– the first success occurs after the kthtrial– the first success occurs on or after the kthtrial– the first success occurs before the kthtrial– the first success occurs on or before the kthtrial4• Negative binomial probability distribution:Suppose that repeated Bernoulli trials are performed until r suc-cesses occur. The number of trials required X, follows the socalled negative binomial probability distribution.Probability mass function of X is:P (X = x) =x − 1r − 1pr−1(1 − p)x−rp, orP (X = x) =x − 1r − 1pr(1 − p)x−rx = r, r + 1, r + 2, · · ·5• Hypergeometric probability distribution:Select without replacement n from N available items (of whichr are labeled as “hot items”, and N − r are labeled as “colditems”). Let X be the number of hot items among the n.Probability mass function of X:P (X = x) =rxN−rn−xNn6California Super Lotto Plus:7• Poisson probability distribution:The Poisson probability mass function with parameter λ > 0(where λ is the average number of events occur per time, area,volume, etc.) is:P (X = x) =λxe−λx!, x = 0, 1, 2, · · ·8• Multinomial probability distribution:A sequence of n independent experiments is performed and eachexperiment can result in one of r possible outcomes with prob-abilities p1, p2, . . . , prwithPri=1pi= 1. Let Xibe the numberof the n experiments that result in outcome i, i = 1, 2, . . . , r.Then,P (X1= n1, X2= n2, . . . , Xr= nr) =n!n1!n2! · · · nr!pn11pn22· · · pnrr9Example:Suppose 20 patients arrive at a hospital on any given day. Assume that 10% of all thepatients of this hospital are emergency cases.a. Find the probability that exactly 5 of the 20 patients are emergency cases.b. Find the probability that none of the 20 patients are emergency cases.c. Find the probability that all 20 patients are emergency cases.d. Find the probability that at least 4 of the 20 patients are emergency cases.e. Find the probability that more than 4 of the 20 patients are emergency cases.f. Find the probability that at most 3 of the 20 patients are emergency cases.g. Find the probability that less than 3 of the 20 patients are emergency cases.h. On average how many patients (from the 20) are expected to be emergency case?Example:Patients arrive at a hospital. Assume that 10% of all the patients of this hospital areemergency cases.a. Find the probability that at any given day the 20thpatient will be the first emer-gency case.b. On average how many patients must arrive at the hospital to find the first emer-gency case?c. Find the probability that the first emergency case will occur after the arrival ofthe 20thpatient.d. Find the probability that the first emergency case will occur on or before thearrival of the 15thpatient.Example:The probability for a child to catch a certain disease is 20%. Find the probability thatthe 12thchild exposed to the disease will be the 3rdto catch it.Example:A basket contains 20 fruits of which 10 are oranges, 8 are apples, and 2 are tangerines.You randomly select 5 and give them to your friend. What is the probability thatamong the 5, your friend will get 2 tangerines?Example:In a bag there are 10 green and 5 white chips. You randomly select chips (one at atime) with replacement until a green is obtained. Your friend does the same. Find thevariance of the sum of the number of trials needed until the first green is obtained byyou and your friend.Example:The number of industrial accidents at a particular plant is found to average 3 permonth. Find the probability that 8 accidents will occur at any given month.10Review problemsProblem 1Suppose X ∼ b(n, p).a. The calculation of binomial probabilities can be computed by means of the followingrecursion formula. Verify this formula.P (X = x + 1) =p(n − x)(x + 1)(1 − p)P (X = x)b. Let X ∼ b(8, 0.25). Use the above result to calculate P (X = 1), and P (X = 2). Youare given that P (X = 0) = 0.1001.Problem 2The amount of flour used per week by a bakery is a random variable X having an exponentialdistribution with mean equal to 4 tons. The cost of the flour per week is given by Y = 3X2+1.a. Find the median of X.b. Find the 20thpercentile of the distribution of X.c. What is the variance of X?d. Find P (X > 6/X > 2).e. What is the expected cost?Problem 3Answer the folowing questions:a. If the probabilities of having a male or female offspring are both 0.50, find the proba-bility that a family’s fifth child is their second son.b. Suppose the probability that a car will have a flat tire while driving on the 405 freewayis 0.0004. What is the probability that of 10000 cars driving on the 405 freeway fewerthan 3 will have a flat tire. Use the Poisson approximation to binomial for fastercalculations.c. A doctor knows from experience that 15% of the patients who are given a certainmedicine will have udesirable side effects. What is the probability