Statistics and Probability Answers

1)Suppose E(x)=3 and Var(x)=4. Compute E(Y) and Var(Y) when

Y=3X+2

Y=-5X+4

2)Suppose that a large number of containers of candy are made up of two types, say A and B. Type A contains 70% sweet and 30% sour ones while for type those percent’s are reversed. Furthermore, suppose that 60% of all candy jars are of type A, while the remain are of type B. If you are given a jar of unknown type and draw one pieces of candy;

What is  the probability that it is type A given that it sweet?

What is the probability that it is type B given that it sweet?

If X is a random variable such that E[X] = 10 and σ _X = 3, can X have a negative bino-

mial distribution NB(r; p) in which X represents the number of failures preceding the r^th

success? Explain.

5) There are three families each having four children, 2 boys and 2 girls; 3boys and 1 girl; and 1 boy and 3 girls. A child from each family is invited to 

a party. Find the probability  

(i) the only girls turn up for the party?  

(ii) the two girls and one boy turn up for the party? 

Q. 1. joint pdf of random variables ‘X’ and ‘Y’ is

F(x,y)=1/2, 0 <x<y, 0<y<2

Find

a. The marginal pdfs, f_X(x) and f_Y(y).

b. The conditional pdfs, f_X/Y(x/y) and f_Y/X(y/x)

c. The E(X/Y=1)

d. Are ‘X’ and ‘Y’ statistically The independent?

Marks of six students are tabulated below:

Name : Raj Anil Amit Om Rita Renu

Marks : 54 50 52 48 50 52

From the population, tabulated above, you are supposed to choose a sample of size two.

(a) Determine, how many samples of size two are possible

(b) Construct sampling distribution of means by taking samples of size 2 and organize the data.

A survey of CPAs across the US found that the average net income for sole proprietor CPAs is not more than $74,924. Because this survey is more than 15 years old, an accounting researcher who wanted to test this finding took a sample of 112 sole proprietor accountants and found the net income to be $78,695 with a population standard deviation of $14,530. If you want to test at the 5% significance level if the survey findings still hold true, what are your null and alternative hyposthesis?

Ho : μ ≤ 74,924; Ha : μ > 74,924

Ho : μ > 74,924; Ha : μ < 74,924

Ho : x̄ < 74,924; Ha : x̄ > 74,924    

Ho : μ = 74,924; Ha : μ≠74,924

In a study of the life expectancy of 400 people in a certain geographic region, the mean age at death was 70 years, and the standard deviation was 5.1 years.

If a sample of 50 people from this region is selected, what is the probability that the mean life expectancy will be more than 90 years?

If is estimated that 30% of all students who fall under the Business Economics and Management department at Richfield have drivers licence. What is the probability of this in a sample of 10 students

Exactly 4 students have drivers license

Q. 1. The joint pdf of random variables ‘X’ and ‘Y’ is

Find

a. The marginal pdfs, f_X(x) and f_Y(y).

b. The conditional pdfs, f_X/Y(x/y) and f_Y/X(y/x)

c. The E(X/Y=1)

d. Are ‘X’ and ‘Y’ statistically independent?

1.  (6 marks) A random variable, X, measures a European football team’s net goal differ ence (goals scored minus goals conceded) for matches in a semi-professional league. The values X =-1, 0, 1 indicate negative, null or positive goal difference, respectively. X is thought to follow the probability distribution below:

a.      (2 marks) Complete the table below by computing the values of the probability mass function, P (x), and the cumulative distribution function, F (x).                   

b.     (2 marks) Compute the expected value of X by writing an R function to do the calculation (show your R code).