概率统计英文chapter Discrete Random Variables and Probability Distributions.ppt

3 Discrete Random Variables and Probability Distributions 3.1 Random Variables Definition: For a given sample space Ω of some experiment, a random variable is any rule that associates a number with each outcome in Ω. 3.6 The Poisson Probability Distribution Example 3 Suppose the probability of getting a rare, non-contagious disease in a given year is 10-4. Consider a town of size 10,000. Determine the probability that five or more of the townspeople come down with the disease in a given year; over a 100-year period there are two or more years when five or more townspeople come down with the disease. Solution: (a) We use Poisson approximation of the binomial probability distribution μ=10000× 10-4 =1 (b) over a 100-year period there are two or more years when five or more townspeople come down with the disease. Solution: in this question, the probability is binomial probability distribution with p=0.037, n=100 Because we can not find the probability from the binomial distribution table, we use Poisson approximation of the binomial probability distribution λ=100× 0.0037 =0.37 Proposition: If X has a Poisson distribution with parameter λ, then E(X)=V(X)= λ Example 1: A discrete random variable X have Poisson distribution with parameter λ, determine the expectation EX. Solution: the Poisson distribution probability function is Example 2: A discrete random variable X have Poisson distribution with parameter λ, determine the expectation V(X). Solution: the Poisson distribution probability function is Exercise P137 78 A notice is sent to all owners of a certain type of automobile, asking them to bring their cars to a dealer to check for the presence of a particular manufacturing defect. Suppose that only 0.05% of such cars have the defect. Consider a random sample of 10,000 cars. What are the expected value and standard deviation of the number of cars in the sample that have the defect? What is the (approximate) probability that more than 10 sampled cars h