Question

In: Statistics and Probability

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.

  1. 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?

      
  2. 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 of fX. That is, find A and B in the formula below.

    fZ|Y(z|y)=AfX(B).

    A=

    B=

  3. Which of the following is a formula for fZ(z)?

    fZ(z)=
    (Choose all that apply.)

    fZ(z)=∫∞0fY,Z(y,z)dy

    fZ(z)=∫∞0fY,Z(y,z)dz

    fZ(z)=∫∞0fY(y)fZ,Y(z,y)dy

    fZ(z)=∫∞0fY(y)fZ|Y(z|y)dy

    fZ(z)=∫∞0fY(y)fX(yz)dy

    fZ(z)=∫∞0yfY(y)fX(yz)dy

Solutions

Expert Solution


Related Solutions

Let X ∼Exp(1), Y ∼Exp(2) be independent random variables. (a) What is the range of Z...
Let X ∼Exp(1), Y ∼Exp(2) be independent random variables. (a) What is the range of Z := X + Y ? (b) Find the pdf of Z. (c) Find MZ(t). (d) Let U = e Y . What is the range of U? (e) Find the pdf of U|X.
Assume that X, Y, and Z are independent random variables and that each of the random...
Assume that X, Y, and Z are independent random variables and that each of the random variables have a mean of 1. Further, assume σX = 1, σY = 2, and σZ = 3. Find the mean and standard deviation of the following random variables: a. U = X + Y + Z b. R = (X + Y + Z)/3 c. T = 2·X + 5·Y d. What is the correlation between X and Y? e. What is the...
Let X, Y, and Z independent random variables with variance 4 and mean 1. Find the...
Let X, Y, and Z independent random variables with variance 4 and mean 1. Find the correlation coefficient between (X-2YX+1) and (4X+Y)
Let X, Y and Z be independent random variables, each uniformly distributed on the interval (0,1)....
Let X, Y and Z be independent random variables, each uniformly distributed on the interval (0,1). (a) Find the cumulative distribution function of X/Y. (b) Find the cumulative distribution function of XY. (c) Find the mean and variance of XY/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. 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. d) Find the covariance of Z and W.
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 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, 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 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...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT