1. Handout Ch5(1) 實習

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

3. Binomial Distribution
If the random variable X1, …, 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
If X1, …, Xk are independent random variables and if Xi has a binomial distribution with parameters ni and p, then the sum has a binomial distribution with parameters and p.

4. 5.2.10
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?

5. Solution

6. 5.3.6
Suppose that X1 and X2 are independent random variable; that X1 has a binomial distribution with parameters n1 and p; and that X2 has a binomial distribution with parameters n2 and p, where p is the same for both X1 and X2. For any fixed value of k (k=1,2…,n1+n2), determine the conditional distribution of X1 given that X1+X2=k

7. Solution

8. Poisson Distribution
X has a Poisson distribution with mean l if the p.m.f. of X has:

9. Poisson Distribution
The moment generating function

10. Poisson Distribution
If the random variables X1, …, 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
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 = X1 + X2 has a Poisson distribution with mean 9.

11. Poisson Approximation to Binomial Distribution
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,

13. 羅必達法則 當x→a時，函數f(x)及g(x)都趨於零； 在點a的附近鄰域內，f’(x)及g’(x)都存在，且g’(x) ≠0
存在(或為無窮大)， 則
各種形式：0/0，∞/∞，0× ∞， ∞- ∞，00， ∞0，1 ∞

14. 5.4.8
Suppose that X1 and X2 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 X1 given that X1+X2=k

15. Solution

16. 5.4.14
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

17. Solution

18. Geometric Distribution
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.
Let X = number of failures to first success.
f ( x | p ) = pqx for x = 0, 1, 2, …
Let Y = number of trials to first success.
f ( y | p ) = pqy–1

19. The m.g.f of Geometric Distribution
If X1 has a geometric distribution with parameter p, then the m.g.f.
It is known that

20. Negative Binomial Distribution
Let X = number of failures to rth success.
Let Y = number of trials to rth success.

21. The m.g.f of Negative Binomial Distribution
If X1, …, Xr are i.i.d. random variables and if each Xi has a geometric distribution with parameter p, then the sum X Xr has a negative binomial distribution with parameters r and p.
If X has a negative binomial distribution with r and p, then the m.g.f.
Memoryless property of the geometric distribution

22. 5.5.10
Let denote the p.f. of the negative binomial distribution with parameter r and p; and let denote the p.f. of the Poisson distribution with mean Suppose in such a way that the value of rq remains constant and is equal to throughout the process. Show that for each fixed nonnegative integer x,

23. Solution