Let X and Y be two independent random variables. X is a binomial (25,0.4) and Y...

Let X and Y be two independent random variables. X is a binomial (25,0.4) and Y is a uniform (0,6). Let W=2X-Y and Z= 2X+Y.

a) Find the expected value of X, the expected value of Y, the variance of X and the variance of Y.

b) Find the expected value of W.

c) Find the variance of W.

d) Find the covariance of Z and W.

Expert Solution

mean and variance of binomial and uniform

Now few properties of covariance

orchestra answered 2 years ago

Let X and Y be two independent random variables. Assume that X is Negative- Binomial(2, θ)...

Let X and Y be two independent random variables. Assume that X is Negative- Binomial(2, θ) and Y is Negative-Binomial(3, θ) distributed. Let Z be another random variable, Z = X + Y . (a) Find the following probabilities: P(Z = 0), P(Z = 1) and P(Z = 2); (b) Can you guess what is the distribution of Z?

Let X and Y be two independent random variables such that X + Y has the...

Let X and Y be two independent random variables such that X + Y has the same density as X. What is Y?

Let X and Y be two independent random variables, and g : R2 --> R an...

Let X and Y be two independent random variables, and g : R2 --> R an arbitrary bivariate function. 1) Suppose that X and Y are continuous with densities fX and fY . Prove that for any y ? R withfY (y) > 0, the conditional density of the random variable Z = g(X, Y ) given Y = y is the same as the density of the random variable W = g(X, y). 2) Suppose that X and Y...

Let X∼Binomial(n,p) and Y∼Bernoulli(p) be independent random variables. Find the distribution of X+Y using the convolution...

Let X∼Binomial(n,p) and Y∼Bernoulli(p) be independent random variables. Find the distribution of X+Y using the convolution formula

Let X, Y be independent exponential random variables with mean one. Show that X/(X + Y...

Let X, Y be independent exponential random variables with mean one. Show that X/(X + Y ) is uniformly distributed on [0, 1]. (Please solve it with clear explanations so that I can learn it. I will give thumbs up.)

Let X and Y be two independent and identically distributed random variables with expected value 1...

Let X and Y be two independent and identically distributed random variables with expected value 1 and variance 2.56. (i) Find a non-trivial upper bound for P(|X + Y − 2| ≥ 1). 5 MARKS (ii) Now suppose that X and Y are independent and identically distributed N(1, 2.56) random variables. What is P(|X + Y − 2| ≥ 1) exactly? Briefly, state your reasoning. 2 MARKS (iii) Why is the upper bound you obtained in Part (i) so different...

Let X and Y be independent Gaussian(0,1) random variables. Define the random variables R and Θ,...

Let X and Y be independent Gaussian(0,1) random variables. Define the random variables R and Θ, by R2=X2+Y2,Θ = tan−1(Y/X).You can think of X and Y as the real and the imaginary part of a signal. Similarly, R2 is its power, Θ is the phase, and R is the magnitude of that signal. (b) Find the probability density functions of R and Θ. Are R and Θ independent random variables?

Let X and Y be independent positive random variables. Let Z=X/Y. In what follows, all occurrences...

Let X and Y be independent positive random variables. Let Z=X/Y. In what follows, all occurrences of x, y, z are assumed to be positive numbers. Suppose that X and Y are discrete, with known PMFs, pX and pY. Then, pZ|Y(z|y)=pX(?). What is the argument in the place of the question mark? Suppose that X and Y are continuous, with known PDFs, fX and fY. Provide a formula, analogous to the one in part (a), for fZ|Y(z|y) in terms...

Let X, Y be independent random variables with X ∼ Uniform([1, 5]) and Y ∼ Uniform([2,...

Let X, Y be independent random variables with X ∼ Uniform([1, 5]) and Y ∼ Uniform([2, 4]). a) FindP(X<Y). b) FindP(X<Y|Y>3) c) FindP(√Y<X<Y).

Let X,,X, and X, be independent uniform random variables on [0,1] Write Y = X, +X,...

Let X,,X, and X, be independent uniform random variables on [0,1] Write Y = X, +X, and Z = X+ X. a.) Compute E[X,X,X,. (5 points) b.) Compute Var(X). (5 points) c.) Compute and draw a graph of the density function fr. (15 points)

Question