Let Z be a random variable given by Z -+ 4Y where X and Y are two independent random variables with probability density functions fe(x)(u(x)-u(*-10)]
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![Let Z be a random variable given by
Z =+ 4Y
where X and Y are two independent random variables with probability density
functions
f«(x) =tu(x) - u(x = 10)]
fe(x) =[u(x + 10) – u(x – 10)]
a) Find a and b.
b) Find the probability density function of Z.
c) What is the probability that 0 s Z S 60?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a9cc907-be00-4e00-91c4-0f92e3dc220a%2Fe2f06a55-f78b-4087-ae3f-b13f9e8b64bd%2Fv3s0i8_processed.jpeg&w=3840&q=75)
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- The random vector (X,Y) has the following joint probability density function:f(X,Y)(x,y) ={4xye−(x^2+y^2), x >0, y >0, 0, otherwise LetZ=√(X2+Y2) Find the probability density of the random variableZ.Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise Find the constant c, P(Y≥1/2), P(X < 2, Y >1/2), P(X < 1), Determine whether X and Y are independent.Let X and Y be independent uniform random variables on (0, 1). Find their joint density function f (x, y). Use the joint density function to calculate the probability P(X < Y).
- Suppose that X, Y are jointly continuous with joint probability density function f( x, y){ xe^-x(1+y), ifx >0 and y >00, otherwise. (a) Find the marginal density functions of X and Y. (b) Calculate the expectation E[XY]. (c) Calculate the expectation EIX/(1+ Y )1. (e) Determine if the random variables X and Y in this exercise are independent.Let X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.Suppose that two-dimensional continuous random variable (X, Y) has joint probability density function given by f(x,y) = 24xy, x is less than equal to 1 and greater than equal to 0, y is less than equal to 1 and greater than equal to 0, x+y is less than equal to 1 and greater than equal to 0. Check that E(Y) = E[E(Y|X)] and V(Y) = E[V(Y|X)] + V[E(Y|X)].
- Suppose that the random variables X and Y have a joint density function f(x,y).prove that Cov(X,Y)=0 if E(X|Y=y) does not depend on yLet X and Y be two independent random variables, X ∼ Γ(α, λ) and Y ∼ Γ(β, λ). Find the joint probability density function f(Z,W)of the vector (Z, W)(b) Show that Z and W are independent(c) Show that Z ∼ Γ(α + β, λ) and W ∼ B(α, β)Suppose the random variables X and Y have joint probability density function f(x,y) given by: (image)Find: P(X < Y) = fX|Y=y (x)
- A zero-mean stationary Gaussian random process X(t) has power spectral density S_x(f). Determine the probability density function of a random variable obtained by observing the process X(t) at some time t_kLet X be a continuous random variable with density function f(x) = {2x if x ∈ [0,1] {0 otherwise Compute E[X] and E(X2).Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE Cov (X,Y) = E[XY] - E[X] E[Y]