University of California, Los AngelesDepartment of StatisticsStatistics 13 Instructor: Nicolas ChristouProbability modelsSome special probability distributions• Bernoulli random variableIt is a variable that has 2 possible outcomes: “success”, or “failure”. Success occurswith probability p and failure with probability 1 − p.1• Geometric probability distributionSuppose that repeated independent Bernoulli trials each one having probability ofsuccess p are to be performed. Let X be the number of trials needed until the firstsuccess occurs. We say that X follows the geometric probability distribution withparameter p.Probability model: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 =q1−pp2Example: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?2• More on geometric probability distribution · · ·Repeated Bernoulli trials are performed until the first success occurs. Find the prob-ability 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 kthtrial3Geometric distribution using the Statistics Online ComputationalResource (SOCR)X follows the geometric probability distribution with parameter p = 0.3.a. Write a probability statement in terms of X for the unshaded area of the firstapplet.b. Write a probability statement in terms of X for the unshaded area of the secondapplet.c. Verify the probabilities in both applets using the formulas P (X > k) = (1 − p)k,etc.4X follows the geometric probability distribution with parameter p = 0.3.a. Write a probability statement in terms of X for the unshaded area of the firstapplet.b. Write a probability statement in terms of X for the unshaded area of the secondapplet.c. Verify the probabilities in both applets using the formulas P (X > k) = (1 − p)k,etc.5X follows the geometric probability distribution with parameter p = 0.3.a. Write a probability statement in terms of X for the shaded area of the first applet.b. You want to compute P (3 ≤ X < 9). Compute this probability using the formu-las, show it on the second applet, and write the Left Cut Off and Right Cut Offpoints.6• Binomial probability distributionSuppose that n independent Bernoulli trials each one having probability of success pare to be performed. Let X be the number of successes among the n trials. We saythat X follows the binomial probability distribution with parameters n, p.Probability model: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 =qnp(1 − p7Example: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?8• Hypergeometric probability distributionSelect without replacement n from N available items (of which r are labeled as “hotitems”, and N − r are labeled as “cold items”). Let X be the number of hot itemsamong the n.Probability mass function of X:P (X = x) =rxN−rn−xNnExpected value of X: E(X) = rnNVariance of X: σ2=nr(N −r)(N −n)N2(N−1)= nrNN−rNN−nN−1Standard deviation of X: σ =rnr(N −r)(N −n)N2(N−1)9California Super Lotto Plus:10Discrete probability distributions - examplesExample 1:Find the probability that 3 out of 8 plants will survive a frost, given that any such plant will survive a frostwith probability of 0.30. Also, find the probability that at least one out of 8 will survive a frost. What isthe expected value and standard deviation of the number of plants that survive the frost?Example 2:If the probabilities of having a male or female offspring are both 0.50, find the probabilities that a family’sfifth child is their first son.Example 3:A complex electronic system is built with a certain number of backup components in its subsystem. Onesubsystem has 4 identical components, each with probability of 0.20 of failing in less than 1000 hours. Thesubsystem will operate if at least 2 of the 4 components are operating. Assume the components operateindependently.a. Find the probability that exactly 2 of the 4 components last longer than 1000 hours.b. Find the probability that the subsystem operates longer than 1000 hours.Example 4:Suppose that 30% of the applicants for a certain industrial job have advanced training in computer program-ming. Applicants are interviewd sequentially and are selected at random from the pool. Find the probabilitythat the first applicant having advanced training in computer programming is found on the fifth interview.Example 5:Refer to the previous exercise: What is the expected number of applicants who need to be interviewed inorder to find the first one with advanced training in computer programming?Example 6:A missile protection system consists of n radar sets operating independently, each with probability 0.90 ofdetecting a missile entering a zone.a. If n = 5 and a missile enters the zone what is the probability that exactly 4 radar sets detect themissile? At least one?b. How large must n be if we require that the probability of detecting a missile that enters the zone be0.999?Example 7:Construct a probability histogram for the binomial probability distribution for each one of the following:n = 5, p = 0.1, n = 5, p = 0.5, n = 5, p = 0.9. What do you observe? Explain.Example 8:On a population of consumers, 60% prefer a certain
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