Let X ~ Unif(0,1). Find a function of X that has CDF F(x) = 1 ̶
x ̶ p for p > 0 (this is the Pareto distribution).
Let X ~ Unif(0,1). Find a function of X that has CDF F(x) = 1 ̶ x ̶ p for p > 0 (this is the P...
Let X have the pdf defined for 0<x<2. Let Y~Unif(0,1). Suppose X and Y are independent. Find the distribution of X-Y. fx() =
(a) Let X be a continuous random variable with the cdf F(x) and pdf f(.1). Find the cdf and pdf of |X|. (b) Let Z ~ N(0,1), find the cdf and pdf of |Z| (express the cdf using ” (-), the cdf of Z; give the explicit formula for the pdf).
a) Let X-Unif(0,1). Derive the pdf of Y =-ln(1-X) Remember to provide its support. Let X-N(1,02). Derive the pdf of Y-ex and remember to provide its support. b) Hint for both parts: First work out the cdf of Y, and then use it to find the density of Y.
Suppose that U~Unif[0,1]. Let . Find the probability density function of Y.
Let X~UNIF(0,1), and Y=-lnX. Then what is the density function of Y where nonzero?
Let X be a random variable with the following cumulative distribution function (CDF): y<0 (a) What's P(X < 2)? (b) What's P(X > 2)? c)What's P(0.5 X < 2.5)? (d) What's P(X 1)? (e) Let q be a number such that F(0.6. What's q?
Suppose that X ~ unif(0,1). Find the distribution of Y = (1 – X)-B – 1 for some fixed B> 0. (Name it!)
1. A certain continuous distribution has cumulative distribution function (CDF) given by F(x) 0, r<0 where θ is an unknown parameter, θ > 0. Let X, be the sample mean and X(n)max(Xi, X2,,Xn). (i) Show that θ¡n-(1 + )Xn ls an unbiased estimator of θ. Find its mean square error and check whether θ¡r, is consistent for θ. (i) Show that nX(n) is a consistent estimator of o (ii) Assume n > 1 and find MSE's of 02n, and compare...
Let X ~ N(0, 1), and let Z ~ Unif{-1, 1} (i.e. P(Z = -1) = P(Z = 1) = 1/2) be independent of X. Let Y = ZX. What is the distribution of Y? Show that X and Y are uncorrelated. Are X and Y independent?
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an Suppose f is a continuous and differentiable function on...