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# Probability & Statistics I MTH 445

Marshall

GPA 3.51

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This 0 page Class Notes was uploaded by Pansy Cronin on Sunday November 1, 2015. The Class Notes belongs to MTH 445 at Marshall University taught by Staff in Fall. Since its upload, it has received 236 views. For similar materials see /class/233251/mth-445-marshall-university in Mathematics (M) at Marshall University.

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Date Created: 11/01/15

11052008 Statistics 445545 STUDY SHEET for MIDTERM Wednesday November 12 The test will cover Ch 3 sections 34 7 39 310 Ch 4 sections 41 7 410 Ch 5 section 52 The best way to prepare to the midterm is to read the book and to do the homework exercises Please take time to go over the material 1 Chapter 3 Some probability distributions 1 3833 e V Discrete Random Variables Binomial probability distribution Table 1 Geometric probability distribution Negative binomial probability distribution Hypergeometric probability distribution approximation of hypergeometric distribution by a binomial distribution Poisson probability distribution Table 3 approximation of a binomial distribution by a Poisson distribution 2 Moments and momentgenerating functions for discrete probability distributions 3 Tchebysheff s theorem II Chapter 4 1 9 4 5 111 Chapter 5 1 Continuous Random Variables Probability Distribution 3 b 0 Distribution function Probability density function Properties Expected values for continuous random variables Some continuous probability distributions a b c d D V 0 Uniform probability distribution Normal probability distribution Standard normal probability distribution Table 4 Gamma probability distribution Exponential probability distribution the memoryless property of the exponential distribution Chisquare probability distribution degrees of freedom Table 6 Beta probability distribution Moments and momentgenerating function for continuous probability distributions Tchebysheff s theorem Multivariate Probability Distributions The joint probability function of two random variables 2 The joint distribution function of two random variables 3 The joint probability density function Problems 1 4 0 gt1 When a white ower is crossed with a red ower the resulting ower has a 25 chance of being white Suppose that 20 such owers are independently produced by crossbreeding a Find the mean and variance for the number of white owers produced b Find the probability that exactly 5 white owers are produced c Find the probability that at most 3 white owers are produced A personal officer is interviewing job applicants Each applicant has a probability of 02 ofbeing bilingual a Find the probability that among 25 applicants at least 5 are bilingual b Find the probability that exactly 5 applicants must be interviewed in order to nd one who is bilingual The number of people entering the intensive care unit at a hospital on any single day possesses a Poisson distribution with a mean equal to five persons per day a What is the probability that the number of people entering the intensive care unit on a particular day is equal to 2 Is at most 2 b Is it likely that the number of people entering the intensive care unit on a particular day will exceed 10 Explain If Y has a Poisson distribution with parameter 9 show that EY74 Find the distribution mean and variance of the random variable that have er ma 2 25 as its moment generating function Let mt Eet 29quot gem is the moment generating function for a random variable Y Find 21 EO b VO c Distribution of Y Derive the moment generating function for a random variable having a geometric distribution with parameter p For a certain type of soil the number of wireworms per cubic foot has a mean of 100 Assuming that a Poisson distribution of wireworms give an interval that will include at least 59 of the sample values of wireworm counts obtained from a large number 1 cubicfoot samples 0 O N LA 4 VI The length of time required by students to complete a 1hour exam is a random variable X with a density function given by OSx 1 0 otherwise axzl x fx a b c Find the constant c so that f x is a probability density function Find the distribution function of X Graph it Find the probability that a randomly selected student will finish in less than half an hour PX lt 05 d Find the expected value and the stande deviation of X A candy maker produces mints that have a label weight of 204 grams Assume that the distribution of these mints is approximately normal with mean 2137 and stande deviation of0 16 a Find the probability that a randomly selected mint has weight anywhere between 2107 and 2162 grams b If the 2 of the mints with smallest weight is considered to be defective what is the actual weight for a mint to be considered defective If Y has an exponential distribution and PY gt 2 00821 what is a 3 b PYSIJIY s2 If X belongs to x2 distribution with 23 degrees of freedom find a P1485 ltXlt 4418 b Constants a and b such that Pa ltXlt b 095 and PX lt a 0025 c The mean and the variance ofX IfX is N01 a2 show that Y aX b is Nau 1241202 a 0 Find the momentgenerating function of the random variable X whose probability density function is given by x gt 0 e x 0 and use it to find the mean and stande deviation of X otherwise Find PIY 1 5 2039 for the uniform random variable with parameters 01 0 and 92 5 Compare with the corresponding probabilistic statements given by Tchebysheff s theorem and the empirical rule Do homework for problems from Section 52

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