Questions in Section Probability/Statistics: page 2

One plane has 4 independently running engines (2 on each side); each engine has a failure rate of 8%. Another plane has 2 engines (1 on each side, also independent); each with a failure rate of 12%. Let X be the number of running engines for the 4 engine plane and Ybe the number of running engines for the 2 engine plane. The 2 engine plane will stay flying as long as one engine works. The 4 engine plane must have at least one working engine on both sides to fly.

A company has created a new lie detector test for law enforcement to use which uses eye scans to determine if a suspect is lying. They claim there are less false positives because the method is less obtrusive than having sensors placed all over the body. They also claim that the test will tell law enforcement if the suspect lies on one question, multiple times or is completely honest. To test the new machine, a state police unit gets 60 volunteers and gives them each a list of 20 questions they will be asked by the machine operator. One third of the group will answer all questions honestly, one third will answer all but question 6 honestly and the rest will randomly lie on at least 5 questions (the people operating and evaluating the machine do not know which person is from which group). 14 of the 20 people who lied multiple times were correctly evaluated, 10 of the 20 who lied on question 6 were correctly evaluated and 8 of the 20 who did not lie were incorrectly evaluated as being deceitful. Studies have shown that about 12% of all people given lie detector tests will lie multiple times and 60% will be completely honest. Assume that these volunteers can be considered a random sample and their results can be generalized for the whole population of people given lie detector tests. Then find the following probabilities.

Let the random variable X have the following pdf: f(x)={(12/5 (1-x)^2 if 0<=x<0.5),(4/15 if 0.5<=x<1),(2/25 (4-x) if 1<=x<2):}.

1. Graph the PDF;
2. Find and graph the CDF of X.

A multiple-choice test has 8 questions each with 4 responses, one of which is correct. The lowest passing grade is 5. Find the probability of obtaining this grade by randomly guessing.

A basketball player misses 20% of his foul shots. If he shoots the ball 7 times, find the probability that he misses

1. exactly 3 of those shots;
2. all 7 those shots. Round this answer to five decimal places.

In testing a new drug, researchers found that 5% of all patients using it will have a mild side effect. A random sample of 10 patients using the drug is selected.

1. make a binomial probability distribution and use it to find the following:
2. probability that none of the patients will have this mild side effect.
3. probability that exactly 2 of the patients will have this mild side effect.
4. probability that at most 3 of the patients will have this mild side effect.

Suppose you buy one ticket at $3 for a raffle that contains 400 tickets, where there is one$250 prize and two \$125 prizes being awarded. What is your expected net gain?

Let X be a random variable with probability density function given by f(x)={(2(1-x) if 0<=x<=1),(0 quad otherwise):}.

1. Find the density function of Y=X^2.
2. Find the mean and variance of Y.

Suppose that B and D are independent exponential random variables with parameters \beta_i and \mu_i respectively.

1. Show that P(B<D)=(\beta_i)/(\beta_i+\mu_i).
2. Show that min(B,D) is exponential random variable with parameter \beta_i+\mu_i.

Your friend flips a fair coin repeatedly until the first head occurs. Let X be the number of flips required. You want to determine how many flips were needed in an experiment. You are allowed to ask a series of yes-no questions of the following form: you give your friend a set of integers and your friend answers "yes" if the number is in set and "no" otherwise.