Question

Class sizes at a certain high school have the following probability distribution. Let X = the class size. Assume the values of X below are the only possible class sizes. X has the following probability mass function:

x	10	20	30	40	50
p(x)	.053	.291	.322	.175	.159

a. If F is the cumulative distribution function of X, then evaluate F(30) (i.e. P(X ≤30)  
b. Calculate the probability that X ≥35  
c. Calculate the expected value of X.  
d. Calculate the expected value ofX²

e. Calculate the standard deviation of X.  

f. The school principal has a great idea. If a class has X students in it, then the teacher gets a bonus of 30* squart root of x.What is the average(expected) teacher bonus in the school

g.If we randomly select classes one by one with replacement until we find a class with at least 30 students. What is the probability that we had to select at least 3 classes?

2.On average, 2 bicycle accidents occur per month in Justice Square. These accidents occur randomly in time and are independent from one another. Thus, we can consider the number, X, of bicycle accidents occurring per month in Justice Square as a Poisson random variable with parameter λ. Let Y = the number bicycle accidents occurring in Justice Square in a year..

a. What is the value of λ?  
b. What is the variance of X?  
c. What is the probability that X = 3?  
d. What is the probability that X > 0?  
e. Y also has a Poisson distribution. What is the parameter λy for Y?  
f. What is variance of Y?  
g. What is the standard deviation of Y?  
h. What is the probability that Y ≤ 20?  

i. What is the variance of 1/2Y?

j. What is the probability of no bicycle accidents during the 1st 3 months of the year?

3.The Mathematics Department is going to have a party and orders drinks (non-alcoholic) to be delivered. The food services department delivers 20 bottles of water and 16 bottles of soda. Professor Newton asks his not too bright Ph.D. student to bring him 6 bottles of the drinks. The student randomly selects 6 bottles from the 36 delivered bottles. Let X = the number of selected bottles which are water. Let Y = the number of selected bottles which are soda.

a. How many ways can 6 out of 36 bottles be selected? (order does not matter)?  
b. How many ways can 6 out of 20 bottles of water be selected? (order does not matter)?  
c. What is the probability that X=6?  
d. What is the probability that X = 5?  
e. What is the probability that Y > 3?  
f. What is the expected value of X?  
g. What is the standard deviation of X?

4.Each time Lebron tries a free throw he has a 75% chance of making it. Each time Michael tries a free throw he has an 82% chance of making it. There is independence between throws Lebron and Michael each try 4 free throws. Of course, the number of free throws Michael makes is independent from the number that Lebron makes. Let X = the number of free throws Lebron makes and let Y = the number of free throws Michael makes.

a. What is the expected value of X?  
b. What is the standard deviation of X?  
c. What is the probability that X = 3?  
d. What is the probability that X < 3?  
e. What is the probability that Y = 3?  
f. What is the probability that X = 3 and Y = 3?  
g. What is the expected value of Y?  
h. What is the expected value of X + Y?  
i. What is the probability that X = Y?  
j. What is the probability that X + Y = 8?  
k. If Lebron keeps trying free throws, what is the probability that he makes all 10 of his 1st 10 tries.?

5.Seatbelts is a built-in R dataset. It is a time series giving data on the monthly totals of car drivers in Great Britain killed or seriously injured Jan 1969 to Dec 1984. There are 192 measurements. We are interested in some descriptive statistics related to the columns front and rear We can access this data directly by using the R assignments x <- as.vector(Seatbelts[,"front"]) and y<-as.vector(Seatbelts[,"rear"])
(In R use ?Seatbelts for info on this dataset.) Using R make the assignments defining x and y as above. If you type length(x) at the R command prompt and press Enter or Return you should get 192. If you type x at the command prompt (>) and press Enter or Return you will see all of the values of x.

a.Calculate the sample 10/192.trimmed mean of y.

b. How many outliers does an R boxplot of x show?  
c. How many outliers does an R boxplot of y show?  
d. The assignment r <- x/y gives the month by month ratio of front seat deaths to rear seat deaths. It calculates element by element. Calculate the maximum value of r.  
e. Calculate the interquartile range of r.  
f. Calculate the range of r.(one number)  
g. Calculate the sample mean of x.  
h. Calculate the sample variance of x.  
i. Calculate the sample standard deviation of x.  
j. What proportion of the x values are more than 2.0 sample standard deviations away from the sample mean of x?  
k. What number do you get if you add up the square roots of the values in x?

6.Apple is selling 3 new smartphones, The iPhone Xs, the iPhone Xs Max, and the iPhone Xr. 35% of new iPhone sales are for the Xs, 18% are for the Xs Max, and 47% are for the Xr. Due to slightly different technologies, the cameras are somewhat different from one another. 84% of iPhone Xs users are highly pleased with the camera. 88% of iPhone Xs Max users are highly pleased with the camera. 68% of iPhone Xr users are highly pleased with the camera. Call someone who has one of these phones a New iPhone user. Suppose we randomly select a New iPhone user Let L be the event that the user is pleased with the camera. Let S be the event that the user has an iPhone Xs. Let M be the event that the user has an iPhone Xs Max. Let R be the event that the user has an iPhone Xr. Let RC be the complement of event R

a. Calculate P(RC)  
b. Calculate the conditional probability P(L|R)  
c. Calculate the probability P(L)  
d. Calculate the probability that the selected user has an iPhone Xr and is highly pleased with the camera.  
e. Calculate the conditional probability that the selected user has an iPhone Xr given that the user likes the camera  
f. Calculate the conditional probability that the selected user has an iPhone Xr given that the user does not like the camera.

7.Let A and B be two events. Suppose P(A) = .7, P(B) = .6 and P(A ∩ B) = .4. Suppose A and B are the complements of A and B respectively.
a. Find the probability of A ∪ B.  
b. Find the conditional probability P(A | B).  
c. Find the conditional probability P(B | A) i.e. the probability of B given A.  
d. Find the probability of (A ∪ B)'.  

e. Find the probability of A' ∩ B'.  
f. Find P(A ∩ B').

8.On a multiple choice test, a student guessing has a 20% (1 in 5) of getting an answer correct. Suppose the student guesses on question after question. Let X = the number of incorrect answers he gets before he gets his 1st correct answer. Let Y = the number of incorrect answers he gets before he gets his 3rd correct answer.

a.What is the expected value of X  
b. What is the variance of X?  
c. What is the probability that X > 10?  
d. What is the probability that X > 7?  
e. What is the probability that Y > 20?  
f. What is the probability that X > 10 given that X > 7?  
g. What is the probability that X > 2?  
h. What is the probability he gets 0 correct in his 1st 8 answers?

Answer 1

#2.
a)
lambda = 2

b)
Variance = lambda = 2

c)
Poisson's Distribution Formula:
P(X = x) = (e^-λ) (λ^x) / x!

P(X=3) = (e^-2) (2^3) / 3! = 0.1804

d)
P(X > 0) = 1 - P(X=0) = 1 - (e^-2) (2^0) / 0! = 0.8647

e)
lambda for Y = 2*12 = 24

f)
variance of Y = 24

g)
sd of Y = sqrt(24) = 4.8990

h)
P(Y <= 20) = 0.2426
=POISSON.DIST(20,24,TRUE)

#7.
P(A) = 0.7, P(B) = 0.6 and P(A and B) = 0.4

a)
P(A or B) = P(A) + P(B) - P(A and B)
= 0.7 + 0.6 - 0.4
= 0.9

b)
P(A|B) = P(A and B)/P(B) = 0.4/0.6 = 0.6667

c)
P(B|A) = P(A and B)/P(A) = 0.4/0.7 = 4/7 = 0.5714

d)
P(A or B)' = 1- P(A or B) = 1- 0.9 = 0.1

e)
P(A' and B') = P(A or B)' = 0.1

f)
P(A and B') = P(A) - P(A and B) = 0.7 - 0.4 = 0.3