Statistics and Probability Answers

A random vector (X, Y, and 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 coefficient between X and Z and between X and Y.

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

James kept careful records of the fuel efficiency of his car. After the first 100 times he filled up the tank, he found the mean was 23.4 miles per gallon (mpg) with a population standard deviation of 0.9mpg. Compute the 95 percent confidence interval for his mpg.

A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared at least once

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

A candidate for mayor in a large city believes that he appeals to at least 10 per cent more of the educated voters than the uneducated voters. He hires the services of a poll-taking organization, and they find that 62 of 100 educated voters interviewed support the candidate, and 69 of 150 uneducated voters support him at the 0.05 significance level. 

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.

James kept careful records of the fuel efficiency of his car. After the first 100 times he filled up the tank, he found the mean was 23.4 miles per gallon (mpg) with a population standard deviation of 0.9mpg. Compute the 95 percent confidence interval for his mpg.