Math Answers

Let X and Y have joint probability distribution function f(x, y) = ( 2x+y /12) , (x, y) = (0, 1); (0, 2); (1, 2); (1, 3) 0, elsewhere

Find

(i) the covariance between X and Y.

(ii) the joint probability generating function of X and Y.

Given that y_1 = e^{x}

y1 =e^x

is a solution of xy’’+(1-2x)y’+(x-1)y=0

xy’’+(1−2x)y’+(x−1)y=0, find its second solution y_2

y2

(0,∞), which is linearly independent from y1

 Let X and Y be two independent random variables having joint probability density function f(x, y) = 1/ 2πσ2 e − (x−µ) 2 σ2 e − (y−µ) 2 σ2 − ∞ < x, y < ∞

Find the moment generating function of Z = X+Y 2 and hence the mean and variance of Z

 A discrete random variable X has probability distribution function f(x) = 12! /x!(12−x)!p x (1 − p) 12−x x = 0, 1, 2, .., 12 0, elsewhere

(i) if p = 0.3, find Pr(X > 3).

(ii) find possible values of p if Var[X] is equal to 1.92. 

 Let A and B be two events defined on a sample space S. If Pr(A)=0.8; Pr(A| B )=0.85 and Pr(A| B^ c )=0.75; determine the probability that neither of the two events occur.

Prove that if U and V are subspaces of Rn

so is U+V

The scores of individual students on a national test have a normal distribution with mean 18.6 and standard deviation 5.9. At Bagabag National High School, 76 students took the test. If the scores at this school have the same distribution as national scores, what are the mean and standard deviation of the sample mean for 76 students?

A population of 1,000 students has an average weekly allowance of μ = 350 Php and standard deviation of σ = 56.13 Php. What is the probability that a random sample of size n = 30 will have an average weekly allowance between 335 and 360 Php?

Calculate the mean and the variance of the discrete random variable x which one values 12 and 3, given that P(1)=10/33, P(2)=1/3 and P(3)=12/33

Given g(x) = (x + 1)(x^2 − x), g: R → R where R is the set of real numbers.

a) Find the domain and range of the function g. (2 marks)

b) Determine whether the function is injective, surjective, and/or bijective. Justify

your answers. (7 marks)