Question

Suppose that the time students wait for a bus can be described by a uniform random variable X, where X is between 0 minutes and 30 minutes. Sketch a picture in both situations to explain your answers. (SHOW YOUR WORK)

A) What is the probability that a student will wait between 5 and 15 minutes for the next bus?

B) What is the probability that a student will have to wait at at most 20 minutes for the next bus?

Suppose that the amounts that students at a college spend on textbooks this semester have a
normal curve with mean = $410 and standard deviation = $100. You must use proper notation here for
full credit. (SHOW YOUR WORK)

C) What is the probability that a randomly selected student spends more than $500 on textbooks
this semester?

D) What is the probability that a randomly selected student spends between $200 and $ 500 on
textbooks this semester?

Suppose the heights of adult males in a population have a normal distribution with mean = 72 inches and standard deviation = 1:5 inches. Two unrelated men will be randomly sampled. Let X = height of the first man and Y = height of the second man. (SHOW YOUR WORK)

E) Consider D = X ?? Y , the difference between the heights of the two men. What type of
distribution will the variable D have?

F) What is the mean value for the distribution of D?

G)Assuming independence between the two men, find the standard deviation of D.

H) Determine the probability that the first man is more than 3 inches taller than the second
man. That is, find P(D > 3).

I) Find the probability that one of the men is at least three inches taller than the other. That is,
find the probability that either D > 3 or D < -3.

Answer 1

We suppose that the time students wait for a bus can be described by a uniform random variable X, where X is between 0 minutes and 30 minutes. Then the pdf of X be

And the sketch be,

A) Then the probability that a student will wait between 5 and 15 minutes for the next bus

=P(5<X<15)

Sketch

B)Now the probability that a student will have to wait at at most 20 minutes for the next bus

Sketch

Now we suppose that the amounts that students at a college spend on textbooks this semester have a
normal curve with mean and standard deviation .

C) Then the probability that a randomly selected student spends more than $500 on textbooks this semester

D) On the other hand the probability that a randomly selected student spends between $200 and $ 500 on textbooks this semester

Now finally we suppose the heights of adult males in a population have a normal distribution with mean inches and standard deviation inches. Two unrelated men will be randomly sampled. Let X = height of the first man and Y = height of the second man.

E) Now we consider D=X-Y, the difference between the heights of the two men. Then by the property of normal distribution that sum or difference of independent normal variables also follows normal distribution, hence

D follows Normal distribution

F) The mean value for the distribution of D

=E(D)

=E(X-Y)=E(X)-E(Y)

=72-72=0

G)Now assuming independence between the two men, the variance of D be,

=Var(D)

=Var(X-Y)

=Var(X)+Var(Y) (since X and Y ae independent hence Cov (X,Y)=0)

Hence the standard deviation of D be,

H) Now the probability that the first man is more than 3 inches taller than the second man. That is,

I) Now the probability that one of the men is at least three inches taller than the other. That is,

As D is symmetrically distributed around 0

Hence the answer...............

Thank you..............