1 Determine the value of c that makes the function fxy c e3x

1- Determine the value of c that makes the function f(x,y)= c e^3x4y a joint probability density function over the range 0<x and 0<y<x. Hence, find the following:

(a) P(Y>3)

(b) P(X<2,Y<2)

(c) P(X<1,Y<2)

(d) P(1<X<2)

(e) E(X)

(f) E(Y)

(g) marginal probability distribution of X

(h)conditional probability distribution of Y given X=1

(i) E(Y|X=1)

(j) conditional probability distribution of X given Y=2

2- Determine the value of K and the correlation and covariance for the joint probability mass function f_x,y(x,y)= Ky+Kx where y=1,2,3 and x=1,2,3.

3- Assume that the weights of individuals are independent and normally distributed with a mean of 76Kg and a standard deviation of 16 Kg. Suppose that 25 people squeeze into an elevator that is designed to hold 1950 Kg.

(a) What is the average and standard deviation of the total weight?

(b) What is the probability that the load (total weight) exceeds the design limit?

Solution

3- Assume that the weights of individuals are independent and normally distributed with a mean of 76Kg and a standard deviation of 16 Kg. Suppose that 25 people squeeze into an elevator that is designed to hold 1950 Kg.

(a) What is the average and standard deviation of the total weight?

average = 76 kg

SD = 16 / srqt (25) = 3.2

(b) What is the probability that the load (total weight) exceeds the design limit?

P( x > 1950 )

P( z > 1950 - 76 / 3.2 )

P( z >585.625) = 0.00000