Consider two continuous random variables Y and Z, and a random variable X that is equal to Y with probability p and to Z with probability 1 − p. Find the PDF of X. Your answer will be in terms of the PDF of Y and Z.

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The first four central moments of a distribution are 0, 16, -36 and 120 respectively. Comment upon the Skewness and Kurtosis of the distribution.

The flight from Greenville to Charlotte can take anywhere from 43 minutes to 68 minutes.  Suppose the random flying times, X, are uniform from 43 to 68 minutes.   On average, how long does it take to fly from Greenville to Charlotte?  μX=μX=  Do not round.

You are given the probability distribution function (PDF) of a continuous random variable X is fX(x). Let Y be a continuous random variable such that Y = aX + b, where a and b are non-zero real constants.1. Find the PDF of Y in terms of  fX , a, and b.2. Let X be an exponential random variable with parameter λ. When will Yalso be an exponential random variable? 3. Let X be a normal random variable with mean μ and variance σ2 . Whenwill Y also be a normal random variable?

Solve: Let X ∼ Exp(λ), Y ∼ Exp(λ), and U ∼ Unif (0, 1). Find the distribution of U (X + Y ).     Please don't handwriting solution

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The variance of Y is σ2y = 4.6875 and the variance of the sum is σ2x + y = 9.1875. You knowthe variance of X=1. Find the covariance, xy. Show work

The hypotenuse, Y, of the isosceles right triangle shown 0 is a random variable having a uniform pdf over the interval [5, 10]. Calculate the expected value of the triangle's area. Do not leave the answer as a function of a. (round to 3 decimal places)

Roll a fair 4-sided die. X is the result. Y is that we toss a fair coin X times and count the number of tails Find the joint distribution for X and Y. Find Cov(X,Y).