Statistics and Probability Answers

Q.4 A random vector (X, Y, Z) has joint density given by 

. f (x, y,z) = k exp [− 1/2 (2x2 − 2xy + y2 + z2 + 2x − 6y) . 

1. Compute k.

2. Compute the expectations P(X), P(Y ) and P(Z).

3. Compute the density of the random vector (X, Z).

4. Compute the correlation coeffificient between X and Z and between X and Y . 

5. Let W = X + Z; compute the probability density of W. 

Q.3 : Let X, Y be two random numbers with joint distribution function 

f(x,y) = Kx fory≤x≤y+1,0≤y≤2, 

= 0 otherwise. 

(a) Compute K. 

(b) Compute the m.d.f. and the expectation of X. 

Q.2 The quickest method to compute the probability that by choosing

by chance 3 pupils, one out of each school, at least one of them wears glasses, is to evaluate the probability that none of them wears glasses. If B is the event that at least one of the 3 pupils wears glasses. 

Q.5 In a Standard Normal Distribution, find:

i. Mean and Standard Deviation

ii. Lower Quartile

iii. Upper quartile

iv. Inter-quartile range

v. Mean Deviation

Q.7: What is the probability that a poker hand of 5 cards contain exactly 2 aces (hypergeometric distribution)?

Q.6 : If an automobile is driven on the average no more than 16000 Km per year, then formulate the null and alternative hypothesis.

: If an automobile is driven on the average no more than 16000 Km per year, then formulate the null and alternative hypothesis

If an automobile is driven on the average no more than 16000 Km per year, then formulate the null and alternative hypothesis.