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- A Troublesome Snowball One winter afternoon, unbeknownst to his mom, a child bring a snowball into the house, lays it on the floor, and then goes to watch T.V. Let W=W(t) be the volume of dirty water that has soaked into the carpet t minutes after the snowball was deposited on the floor. Explain in practical terms what the limiting value of W represents, and tell what has happened physically when this limiting value is reached.E(Yn)? Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0 and limnYn =0a.s.1.9.5. Let a random variable $X$ of the continuous type have a pdf $f(x)$ whose graph is symmetric with respect to $x=c .$ If the mean value of $X$ exists, show that $E(X)=c$Hint: Show that $E(X-c)$ equals zero by writing $E(X-c)$ as the sum of two integrals: one from $-\infty$ to $c$ and the other from $c$ to $\infty .$ In the first, let $y=c-x$ and, in the second, $z=x-c .$ Finally, use the symmetry condition $f(c-y)=f(c+y)$ in the first.
- LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise (a) Show that the moment generating function mX(s) :=E(esX) =λ/(λ−s) for s< λ;Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0
- 5. Suppose Z is a Gaussian random variable with mean 0 and 1. (a) Find the moment generating function of Z2. (b) Find the density function of Z2.1) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.X1 and X2 are two discrete random variables, while the X1 random variable takes the values x1 = 1, x1 = 2 and x1 = 3, while the X2 random variable takes the values x2 = 10, x2 = 20 and x2 = 30. The combined probability mass function of the random variables X1 and X2 (pX1, X2 (x1, x2)) is given in the table below a) Find the marginal probability mass function (pX1 (X1)) of the random variable X1.b) Find the marginal probability mass function (pX2 (X2)) of the random variable X2.c) Find the expected value of the random variable X1.d) Find the expected value of the random variable X2.e) Find the variance of the random variable X1.f) Find the variance of the random variable X2.g) pX1 | X2 (x1 | x2 = 10) Find the mass function of the given conditional probability.h) pX2 | X1 (x2 | x1 = 2) Find the mass function of the given conditional probability.i) Are the random variables X1 and X2 independent? Show it. The combined probability mass function of the random variables X1 and X2 is below
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