2.2 Distribution Functions
87 b. Suppose a count of the total number of heads X and the total number of tails
Y is made after each toss. What is the probability that X always exceeds Y ?
c. What is the probability, after 4 tosses, that X is even if we know that Y ≥ 1? 2. A single expensive electronic part is to be manufactured, but the manufacture of
a successful part is not guaranteed. The first attempt costs 100 and has a 0.7 probability of success. Each attempt thereafter costs 60 and has a 0.9 probability
of success. The outcomes of various attempts are independent, but at most 3 attempts can be made at successful manufacture. The finished part sells for 500.
Find the probability distribution for N , the net profit.
3. An automobile dealer has found that X, the number of cars customers buy each
week, follows the probability distribution f .x =
8 :
k x
2
x ;
x = 1; 2; 3; 4 0;
otherwise.
a. Find k. b. Find the probability the dealer sells at least 2 cars in a week.
c. Find F.x, the cumulative distribution function. 4. Job interviews last one-half hour. The interviewer knows that the probability an
applicant is qualified for the job is 0.8. The first person interviewed who is qualified is selected for the job. If the qualifications of any one applicant are independent
of the qualifications of any other applicant, what is the probability that 2 hours is sufficient time to select a person for the job?
5. Verify the probability distribution for the sum on 3 fair dice as given in Example
2.1.3.
6. a. Since
1 2
+
1 2
5
= 1 and since each term in the binomial expansion of
1 2
+
1 2
5
is greater than 0, it follows that the individual terms in the binomial expansion are probabilities. Suggest an experiment and a sample space for
which these terms represent probabilities of the sample points.
b. Answer part a for . p + q
n
; q = 1 − p; 0 ≤ p ≤ 1:
7. Two loaded dice are tossed. Each die is loaded so the probability that a face, i,
appears, is proportional to 7 − i. Find the probability distribution for the sum that appears. Draw a graph of the probability distribution function.
8. Suppose that X is a random variable giving the number of tosses necessary for a
fair coin to turn up heads. Find the probability that X is even.
88
Chapter 2 Discrete Random Variables and Probability Distributions
Kategori