Chapter 2 exercises - random variables and probability distributions môn Xác suất thống kê| Trường Đại học Ngoại Thương

Chapter 2. Random Variable
1. There are 3 white balls and 2 black balls. Pick each ball at random until you get a white ball. Find the
probability distribution of the number of balls drawn.
2. The probability that a person hits the target is 0.8. He was given each bullet to shoot until he hit the target.
Find the probability distribution of the missed bullet.
3. The probability distribution table of the random variable X is given as follows: X -5 2 3 4 P 0,4 0,3 0,1 0,2 Find E(X); V(X) và σ X
4. At a shop selling Honda motorcycles, the number of motorcycles X sold weekly with the probability
distribution table is as follows: X 0 1 2 3 4 5 6 7 8 9 10 11 P
0,05 0,12 0,17 0,08 0,12 0,20 0,07 0,02 0,07 0,02 0,03 0,05
a) Find the average number of motorbikes sold per week.
b) Find the variance and standard deviation of the number of motorbikes sold per week and interpret the results obtained.
5. Let X1, X2, X3 be independent random variables and have their probability distribution table as follows: X1 0 1 X2 1 2 X3 0 2 P 0.6 0.4 P 0.4 0.6 P 0.8 0.2 X X+ + X Let 1 2 3 X =. Find E( ) X và V( ) X 3
6. Let X and Y are random variables with E(X)= V(X)= 3; E(Y)= V(Y)= 2
Find E(Z) and V(Z) with Z= (3X – 2Y)/5 Z − E( ) Z Find E(T) with T= V( ) Z
7. The discrete random variable X takes on three possible values, x1 = 4 with probability P1 = 0.5; x2 = 0.6
with probability P2 = 0.3 and x3 with probability P3. Find x3 and P3 knowing E(X) = 8.
8. Discrete Random Variable X takes on three possible values: x1= -1, x2=0, x3= 1. Find the corresponding
probabilities knowing that E(X)= 0.1 and E(X2) = 0.9.
9. Experience shows that the quantity of a product purchased by a customer has the following probability distribution: Sonlam Product number 0 1 2 3
The corresponding probabilities 0,5 0,1 0,2 0,2
a) If each product is sold for 110,000 VND and the salesperson gets 10% of the sales, what is the average
amount of commission the salesperson gets from each customer.
b) Find the variance of that commission and state the significance of the results obtained.
10. There are 5,000 people testing their blood for malaria parasites. The local prevalence rate is 10%. The test can be done in two ways: Method 1: Test each person
Method 2: Take blood from 10 people mixed for one test. If the test result is negative (sterility), then
through 10 people no one has the disease. If the test result is positive (with the same), it means that at least 1
in 10 people has the disease. At that time, 10 more odd tests had to be done to detect specific diseases.
Ask which way is more profitable?
11. The annual demand for commodity A is a continuous random variable X with the following probability
density function (unit: thousand products) − ∀ ∈ f (x ) k (30 ) x x (0,30) = ∀ ∉ x 0 (0,30) a) Find k.
b) Find the probability that the demand for that commodity does not exceed 12,000 products per year.
c) Find the average annual demand for that commodity.
12. Waiting time for customers to buy goods is a continuous random variable with the following probability
distribution function (in minutes) 0 if x ≤ 0 3 2 F (x) = 2 − x x 3 x x+ 2< if 0 ≤ 1 1 if x > 1
a) Find the average queuing time
b) Find the probability that out of 3 people in line, no more than 2 people have to wait more than 0.5 minutes
13. Let a random variable X has CDF given by: 0 ∀ x≤ 2 F (x ) 1 = x −1 ∀2< x≤ 4 2 1 ∀ x> 4 . 2 Sonlam
14. Find the probability that in the outcome of the trial X will receive the value: a. Less than 3; b. Belong to [2;3)
15. According to the census, a person in their 40s will live another year by 0.995. A company sells
insurance to people of that age for 10 thousand dong and in case that person dies, it will compensate 1
million dong. Ask the company's average profit from selling an insurance card.
16. The daily price of sugar on the world market for sugar (unit: USD/foot) has the following probability distribution table: X 0,78 0,79 0,8 0,81 0,82 0,83 PX 0,05 0,1 0,25 0,4 0,15 0,05
a) Find the probability that the price of sugar will one day reach at least 0.8 USD/foot.
b) Find the probability that the price of sugar will one day be lower than $0.82/foot.
c) Assume the daily prices of the sugar are independent. Find the probability that for two consecutive days
the price of sugar is higher than 0.8USD/feet.
17. Profit X is obtained when investing 50 million VND in a project with the following probability
distribution table (in million VND) X -2 -1 0 1 2 3 Px 0,1 0,1 0,2 0,2 0,3 0,1
a) Find the most likely return on investment in that project.
b) Is the investment in this project effective? Why?
c) How is the risk of this investment measured? Look for that level of risk.
18. Profit earned from 1 million VND invested in company A and company B have the following
probability distribution tables: X -500 -100 100 500 700 A P 0,2 0,3 0,2 0,2 0,1 XA B X -200 50 100 B X P 0,1 0,6 0,3
a) If you intend to invest 10 million dong, what is the expected profit when investing in companies A and B. 3 Sonlam
b) If the coefficient of variation is used as a measure of the riskiness of an investment, which company is riskier to invest in?
19. A firm hires an attorney in a case with two payment options as follows:
Option 1: Pay 5 million regardless of winning or losing the lawsuit.
Option 2: Pay 100,000 VND if the lawsuit is lost and 15 million VND if the case is won. The lawyer has
chosen option 2. So according to the lawyer's assessment, what is the minimum possibility of winning the company's lawsuit?
20. The daily requirement for a fresh food has the following probability distribution Demand ( kg) 30 31 32 33 34 35 Probabilities 0,15 0,2 0,35 0,15 0,1 0,05
Each kilogram of food is bought for 2.5 thousand and sold for 4 thousand. If they are sold out at the end of
the day, they have to sell them at a discount of 1.5 thousand to sell them all. So how many kg of food must
be ordered daily to make the most profit.
21. The owner of a civil electrical repair shop hires 5 civil electricians to hire 5 electricians to work 40 hours
a week with a salary of 800 thousand/week. To see if another worker needed to be hired, the boss surveyed
demand for X and obtained the following numbers: Demand X Probabilities 180 - 190 0,03 190 - 200 0,09 200 - 210 0,12 210 - 220 0,15 220 - 230 0,22 230 - 240 0,21 240 - 250 0,13 250 - 260 0,05
If every hour the restaurant owner earns 30,000 VND for repairing, should he hire another worker if:
a) Five former workers only agree to work exactly 40 hours/week.
b) Five former workers agree to work a maximum of 5 hours a week each with a salary of 25,000 VND/hour. 4