Statistics and Probability Answers

1. Suppose that the error in the reaction temperature, in ◦C, for a controlled laboratory experiment is a continuous random variable X having the probability density function
f(x)=x2/3 for −1 <x<2,
f(x)=0, elsewhere

(a) Verify that f(x) is a density function
(b) Find P(0 <X≤ 1).

According to the Sleep Foundation, the average night’s sleep is 7 hours. Assume the standard deviation is 0.5 hours and that the probability distribution is normal.
a. What is the probability that a randomly selected person sleeps more than 9 hours?
b. What is the probability that a randomly selected person sleeps 5 hours or less?
c. What is the probability that a randomly selected person sleeps exactly 10 hours?
d. What is the probability that a randomly selected person sleeps more than 7 hours?
e. Doctors suggest getting between 7 and 9 hours of sleep each night. What percentage of the population gets this much sleep?

32. A beach resort buy a policy to insure against loss of revenues due to major storms in the
summer. The policy pays a total of $ 50,000 if there is only one major storms during
the summer, a total of $ 100,000 if there are two major storms, and a total payment
of $ 200,000 if there are more than two major storms. The number of major storms in
one summer is modeled by a Poisson distribution with mean of 0.5 per summer. Find
(a) Find the expected premium for this policy during one summer.
(b) the standard deviation of the cost of providing this insurance for one summer.

The number of typos in a book has a poisson distribution with an average of 5 typos
per 100 pages. What is the probability that you cannot find a typo in 50 pages.

You throw a die repeatedly until you get a 6. What is the probability that you need
to throw more than 20 times to get 6?

(c) (c) When the candidate’s exam is being graded, the grader finds that the candidate
has answered the first five exam questions all wrong. Can the grader conclude that
the candidate is more or less likely to guess a problem right in other problems the candidate’s exam is being graded, the grader finds that the candidate
has answered the first five exam questions all wrong. Can the grader conclude that
the candidate is more or less likely to guess a problem right in other problems

28. A candidate is taking a multiple-choice exam. For each tested in the exam, there are
5 possible choices i.e A, B, C, D, and E. Because the candidate zero knowledge of the
subject, he relies on pure guesswork to answer each question independent of how he
answers any previous questions. Find
(a) the probability that the candidate answers three questions wrong in a row before
he finally answers the fourth question correctly.
(b) let X= the number of problems the candidate answers wrong in a row before he
finally guesses a correct answer.

On a given day, each computer in the lab has at most one crash. There is a 5% chance
that a computer has a crash during the day, independent of the performance of any
other computers in the lab. Find the probability that on any given day, there are;
(a) exactly 3 crashes
(b) at most 3 crashes
(c) at least 3 crashes
(d) more than 3 and less than 6.

At a checkout counter customers arrive at an average of 2 per minute. Find the
probability that
(a) at most 4 will arrive at any given minute.
(b) at least 3 will arrive during an interval of 2 minutes
(c) 5 will arrive in an interval of 3 minutes.

Show that the Geometric distribution r.v satisfies the equation
P(X > i + j|X > i) = P(X > j)