Statistics and Probability Answers

Q(1) [10 Marks] [CLO2,C3] (a) You take an exam that contains 20 multiple-choice questions. Each question has 4 possible options. You know the answer to 10 questions, but you have no idea about the other 10 questions so you choose answers randomly. Your score X on the exam is the total number of correct answers. Find the PMF of X. Find P(X > 15). (b) Packets at a certain node on the internet arrive with a rate of 100 packets per minute. Find the probability that no packets arrive in 6 seconds. Find the probability that 2 or more packets arrive in the first 6 seconds.

50 students live in a dormitory. The parking lot has the capacity for 30 cars. If each student has a car with probability 1 2 (independently from other students), what is the probability that there won’t be enough parking spaces for all the cars.

Draw a scatterplot of the data below (don’t forget to label the x and y axes! Plus include units)

Total number of hours studied (x): 5 25 10 30

QM 3 Final Exam Score (y) 55 85 75 90

Determine the probability when a die is thrown 2 times such that there are no fours and no fives occur?

The home loan department of BRAC Bank Limited sanction a significant number of loans per month. In this month the amount of money requested on home loan applications at a Bank follow a normal distribution with a mean of Tk. 73 lacs and a standard deviation of Tk. 22 lacs. A loan application is received this morning. Find the probability that:

a. The amount requested is Tk. 75 lacs or more?

b. The amount requested is Tk. 45 lacs or less?

c. The amount requested is between Tk. 55 lacs and Tk. 90 lacs?

d. The amount requested is exactly Tk. 65 lacs?

e. The amount requested that would be in the lowest 8%?

f. Also, what is the the median and mode amount?

1.Suppose on average, Nepal experiences 6 earthquakes per year.

a. What is the mean number of earthquakes in Nepal in the first four month of a year?

b. What is the probability that there’ll be 7 earthquakes in Nepal in the next two years?

c. What is the probability that there’ll be at least 9 earthquakes in Nepal in 2021?

2.A TV manufacturing company is planning to launch a new type of product recently. To check the lifespan status of their previous products they conduct a study. And they found that the lifetime of their previous products (plasma TV sets) follows an exponential distribution with a mean of 100,000 hours. Compute the probability a television set:

a. Fails in less than 10,000 hours.

b. Lasts more than 120,000 hours.

c. Fails between 60,000 and 100,000 hours of use.

d. Find the 90th percentile. So, 10 percent of the TV sets last more than what length of time?

1. There are 4 red balls, 6 blue balls and 2 white balls in a bag. Suppose on every single turn, you randomly select a ball, see the color of it and put it back in the bag. You keep doing this repeatedly.

a. What is the probability that you get the first red ball on the 5th turn?

b. How many turns are expected to get one non-white ball?

c. What is the variance of the number of turns required to get one blue ball?

2. There are 4 red balls, 6 blue balls and 2 white balls in a bag. Suppose on every single turn, you randomly select a ball, see the color of it and put it back in the bag. Let’s say, you do this 6 times.

a. What is the probability that you get exactly 3 blue balls after 6 turns?

b. What is the probability that you pick more than 4 blue balls after 6 turns?

c. What is the mean number of red balls picked after 48 turns?

d. What is the standard deviation of the number of white balls picked after 36 turns?

1. A discrete random variable 𝑋 has the following probability mass function

𝑃(𝑋=𝑥)={2𝑘𝑥 𝑥=2,4,6

{𝑘(𝑥+2) 𝑥=8

{ 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

where 𝑘 is a constant

a. Show that 𝑘=1/34

b. Find the exact value of 𝑃(4<𝑥≤8)

c. Find the exact value of 𝑃(2<𝑥<4)

d. What is the expected value of the random variable 𝑋?

e. What is the variance of the random variable 𝑋?

f. Determine 𝑉𝑎𝑟(5−3𝑋)

Given that the random "W" is binomial distribution with "n" trials and success probability "p" in each trial and "P(W=w)=h(w)",show that

(a)."h(w)\/h(w-1)=p(n-w+1)\/(1-p)w,w>0"

(b)."E[W(W-1)]=n(n-1)p"²

(c)."E(1\/W+1)=1-(1-p)""n+1" /(n+1)p

Compute the mean and standard deviation of the population 2,6,8,10,12,14