Statistics and Probability Answers

Given the data below for the actual sale of cases of beer for eight days (Sunday – Sunday)

ANSWER

MONTH

Actual SALE OF CASES OF BEER

Sun

440

Mon

650

Tue

652

Wed

550

Thu

456

Fri

421

Sat

652

Sun

587

Monday

????

(a)​Forecast using a 4 month simple moving average, 3 month weighted moving average ​(using ​weights of 4, 3, and 1, respectively), and an exponential smoothing at alpha = 0.27​​​​​​​​​​​​(9 marks)

(b)​Using the MAD Error Analysis, determine which forecast is preferred.​​(9 marks)​​

(c)​Use the preferred method to forecast Monday of the following Week.​​(2 marks)

Suppose that the occurrence of accidents in a large industrial plant satisfies the three 

assumptions for the Poison process. Suppose also that accidents occur at the rate of ½ per week. Let X be the number of accidents that will occur in a period of the next six weeks.

• A) What is the parameter λ?

• B) Write down the pdf of X.

• C) Find E(X), Var(X) and mgf of X

• D) What is the probability that exactly 3 accidents will occur?

• E) What is the probability that no accidents will occur?

F) If the number of accidents occurring on a highway each day is a poison random variable with parametor "\\lambda"=3  what is the probability that no

accidents occur today?

1.Electric fuses produced by PEM Electric are packaged in boxes of 12 units each. Suppose an inspector randomly selects three of the 12 fuses in a box for testing. If the box contains exactly five defective fuses, what is the probability that the inspector will find exactly one of the three fuses defective?

• Find 𝐸(𝑋) and 𝑉ar(𝑋)

• What is the probability of finding at least 1 defective fuse?

2.There are 20 television sets in a village at a given time. 9 of them are tuned to channel 1,while 11 of them are tuned to channel 2. A sample of size 5 of the 20 sets is randomly selected at this specific time, and it is observed for each of the 5, whether the set is tuned to channel 1 or 2. Let X be the number of sets in the sample that are tuned to channel 1.

• A) What is the probability distribution function of X?

• B) What is the probability that 3 or more TV sets in the sample are tuned to channel 1?

• C) What is the expected value and variance of X?

1.A box of billiard balls is inspected. The inspector draws a ball randomly and independently until he finds a ball with a certain brand. It is known that 10% of all balls have that specific brand.

• a). What is the probability distribution function of 𝑋?

• b). What is 𝐸(𝑋) , 𝑉(𝑋) and 𝑀_𝑋(𝑡) ?

• c). What is the probability that 10 balls will be drawn before finding one ball with the required brand?

2.A fair coin is flipped until a head occurs. What is the probability that less than 3 flips are required; that less than 4 flips are required?

1.....Let X be a random variable with a binomial distribution. Then

E(X) = np

V(X) = npq

M_x(t) = (pe^t + q)ⁿ

Using the direct method, prove the above theorems.

2.....9% of students in the class have their bank balances greater than M500. Suppose that 10 students are selected at random to be interviewed about their bank balances

i) What is the probability that two students will have their bank balances greater than M500?

ii) What is the probability that none will have a bank balance greater than M500?