Handout Ch 5 Review Bernoulli Distribution l A

Handout Ch 5 Review

Bernoulli Distribution l A random variable X has a Bernoulli distribution if Pr(X = 1) = p and Pr(X = 0) = 1– p = q l The p. m. f. of X can be written as l 2 Jia-Ying Chen

Binomial Distribution l l If the random variable X 1, …, Xn form n Bernoulli trials with parameter p, and if , then X has a binomial distribution. The p. m. f. of X can be written as l l 3 If X 1, …, Xk are independent random variables and if Xi has a binomial distribution with parameters ni and p, then the sum has a Jia-Ying Chen binomial distribution with parameters and p.

Example 1 (5. 2. 10) l 4 The probability that each specific child in a given family will inherit a certain disease is p. If it is known that at least one child in a family of n children has inherited the disease, what is the expected number of children in the family who have inherited the disease? Jia-Ying Chen

Solution 5 Jia-Ying Chen

Poisson Distribution l X has a Poisson distribution with mean l if the p. m. f. of X has: l l 6 Jia-Ying Chen

Poisson Distribution l l 7 The moment generating function Jia-Ying Chen

Poisson Distribution 8 l If the random variables X 1, …, Xk are independent and if Xi has a Poisson distribution with mean , then the sum has a Poisson distribution with mean Proof: Let denote the m. g. f. of Xi and denote the m. g. f. of the sum l Example 5. 4. 1: The mean number of customers who visit the store in one hour is 4. 5. What is the probability that at least 12 customers will arrive in a two-hour period? X = X 1 + X 2 has a Poisson distribution with mean 9. Jia-Ying Chen

Poisson Approximation to Binomial Distribution l l When the value of n is large and the value of p is close to 0, the binomial distribution with parameters n and p can be approximated by a Poisson distribution with mean np. Proof: For a binomial distribution with l= np, we have As , then Also, 9 Jia-Ying Chen

Example 2 (5. 4. 8) l 11 Suppose that X 1 and X 2 are independent random variables and that Xi has a Poisson distribution with mean (i=1, 2). For each fixed value of k (k=1, 2, …), determine the conditional distribution of X 1 given that X 1+X 2=k Jia-Ying Chen

Solution 12 Jia-Ying Chen

Example 3 (5. 4. 14) l 13 An airline sells 200 tickets for a certain flight on an airplane that has only 198 seats because, on the average, 1 percent of purchasers of airline tickets do not appear for the departure of their flight. Determine the probability that everyone who appears for the departure of this flight will have a seat Jia-Ying Chen

Solution 14 Jia-Ying Chen

Geometric Distribution l Suppose that the probability of a success is p, and the probability of a failure is q=1 – p. Then these experiments form an infinite sequence of Bernoulli trials with parameter p. l Let X = number of failures to first success. f ( x | p ) = pqx for x = 0, 1, 2, … l Let Y = number of trials to first success. f ( y | p ) = pqy– 1 l 15 Jia-Ying Chen

The m. g. f of Geometric Distribution l If X 1 has a geometric distribution with parameter p, then the m. g. f. It is known that l 16 Jia-Ying Chen

Normal Distribution 19 l There are three reasons why normal distribution is important l Mathematical properties of the normal distribution have simple forms l Many random variables often have distributions that are approximately normal l Central limit theorem tells that many sample functions have distributions which are approximately normal l The p. d. f. of a normal distribution Jia-Ying Chen

極座標 l l 20 直角座標與極座標的轉換 ∫ ∫dxdy=rdrdθ X=r*cosθ，y=r*sinθ Ex: x 2+y 2≦ 1 r (x, y ) Jia-Ying Chen

The m. g. f. of Normal Distribution l 21 Jia-Ying Chen

Properties of Normal Distribution l If the random variables X 1, …, Xk are independent and if Xi has a normal distribution with mean mi and variance si 2, then the sum X 1+. . . + X has a normal distribution with mean m +. . . + m and k 1 k 2. . . 2 variance s 1 + + sk. Proof: 22 l The variable a 1 x 1 +. . . + akxk+ b has a normal distribution with mean a 1 m 1 +. . . + akmk + b and variance a 12 s 12 +. . . + ak 2 sk 2 l Suppose that X 1, …, Xn form a random sample from a normal distribution with mean m and variance s 2 , and let denote the sample mean. Then has a normal distribution with mean m and s 2/n. Jia-Ying Chen

Example 5 (5. 6. 11) l 23 Suppose that a random sample of size n is to be taken from a normal distribution with mean μ and standard deviation 2. Determine the smallest value of n such that Jia-Ying Chen

Solution 24 Jia-Ying Chen

Example 6 l 25 Suppose that the joint p. d. f. of two random variables X and Y is as follows. Show that these two random variable X and Y are independent. Jia-Ying Chen

Solution 26 l Recall that suppose X and Y are random variables that have a continuous joint p. d. f. Then X and Y will be independent if and only if, for and l And l Therefore, X and Y are independent Jia-Ying Chen

Exponential Distribution l A gamma distribution with parameters a = 1 and b is an exponential distribution. l A random variable X has an exponential distribution with parameters b has: l l 27 Memoryless property of exponential distribution Jia-Ying Chen

Life Test l Suppose X 1, …, Xn denote the lifetime of bulb i and form a random sample from an exponential distribution with parameter β. Then the distribution of Y 1=min{X 1, …, Xn} will be an exponential distribution with parameter n β. Proof: l Determine the interval of time Y 2 between the failure of the first bulb and the failure of a second bulb. l l l 28 Y 2 will be equal to the smallest of (n-1) i. i. d. r. v. , so Y 2 has an exponential distribution with parameter (n-1) β. Y 3 will have an exponential distribution with parameter (n-2) β. The final bulb has an exponential distribution with parameter β. Jia-Ying Chen

Physical Meaning of Exponential Distribution 29 l Following the physical meaning of gamma distribution, an exponential distribution is the time required to have for the 1 st event to occur, i. e. , where β is rate of event. l In a Poisson process, both the waiting time until an event occurs and the period of time between any two successive events will have exponential distributions. l In a Poisson process, the waiting time until the nth occurrence with rate b has a gamma distribution with parameters n and b. Jia-Ying Chen

Example 7 (5. 9. 10) l 30 Suppose that an electronic system contains n similar components that function independently of each other and that are connected in series so that the system fails as soon as one of the components fails. Suppose also that the length of life of each component, measured in hours, has an exponential distribution with mean μ. Determine the mean and the variance of the length of time until the system fails. Jia-Ying Chen

Solution 31 Jia-Ying Chen

Example 8 (5. 9. 11) l 32 Suppose that n items are being tested simultaneously, the items are independent, and the length of life of each item has an exponential distribution with parameter β. Determine the expected length of time until three items have failed. Hint: The required value is E(Y 1+Y 2+Y 3) Jia-Ying Chen

Solution 33 Jia-Ying Chen