Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT 1301 PROBABILITY AND STATISTICS I
EXAMPLE CLASS 5
Discrete Uniform Distribution
A random variable X has a discrete uniform[1, N] distribution if P
1, 2, , , where N is a
A Bernoulli random variable X takes on only two values: 0 and 1, with probabilities 1
and p, respectively.
Its pmf is given by P
Suppose that n independent trials, each of which results in a “success” with probability p, are to be performed.
If X represents the number of successes that occur in the n trials, then X is said to be a binomial random
0, 1, , .
variable with parameters (n, p). Its pmf is given by P
A random variable X that takes on one of the values 0, 1, 2 … is said to be a Poisson random variable with
0, if its pmf is given by P
0, 1, .
Consider independent trials, each of which is a success with probability p. If X represents the number of the
first trial that is a success. Its pmf is given by P
1, 2, .
Negative Binomial Distribution
A random variable X which denotes the number of trials needed to amass a total of r successes when each
trial is independently a success with probability p, then X is said to be a negative binomial random variable
with parameters p and r. Its pmf is given by P
If there are 2 types of objects which N are of type I and M are of type II. If a sample of size of n is randomly
chosen, then X, the number of type I selected, has pmf P
C , max ,0 min , N . Find E(X) if X is a random variable with discrete uniform distribution [A, B], where A and B are
. (Hint: ∑
natural numbers with A < B. Hence, show that Var(X) ) Ex.2 A typesetter, on the average, makes one error in every 500 words typeset. A typical page contains 300
words. What is the probability that there will be no more than two errors in five pages? Ex.3 A newsboy purchases papers at 5 dollars and sells them at 6.5 dollars. However, he is not allowed to
return unsold papers. If the daily demand is a binomial random variable with n = 8 and p = 0.5,
approximately how many papers should he purchase so as to maximize his expected profit? Ex.4 An auto parts store has 200 rebuilt starters in stock, of which 4 are defective. In response to a purchase
order, they randomly select 3 starters from their stock. What is the probability that the customer is given
2 defective starters? Ex.5 Let g(x) be a function with ∞
(a) if ~Poisson , then E
(b) if ~negative binomial , X
and ∞ and ∞
, then E 1 1 ∞. Prove that
X E 1 . ...
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