
(1) The random variable Y is said to be obtained from the random variable X by truncation at the point a if X(w) if X(w) S a Write the distribution of Y in terms of the distribution of X. (1) Th...
3. A random variable X is said to have a Cauchy(α, β) distribution if and only if it has PDF function Now, suppose that Xi and X2 are independent Cauchy(0, 1) random variables, and let Y = X1 + X2. Use the transformation technique to find and identify the distribution of Y by first finding the joint distribution of Xi and Y. (Seahin 3 4
Let X and Y be independent random variables. Random variable X has a discrete uniform distribution over the set {1, 3} and Y has a discrete uniform distribution over the set {1, 2, 3}. Let V = X + Y and W = X − Y . (a) Find the PMFs for V and W. (b) Find mV and (c) Find E[V |W >0].
The random variable X is obtained from the following random process. First, you draw from the Uniform(0, 1) distribution. If the value you get is < 0.5, then X is set to equal this value. Otherwise, you draw again (i.e. a second, independent draw from the Uniform(0, 1) distribution), and X is set to this second value. Compute the CDF and the density of X.
Let the random variable X have a uniform distribution on [0,1] and the random variable Y (independent of X) have a uniform distribution on [0,2]. Find P[XY<1].
7. A positive random variable Y is said to be a lognormal random variable, LOGN (u, 0), if In Y ~ N(No?). We assume that Y, LOGN (1,0%), i = 1,..., n are independent. [5] (a) Find the distribution of T = 11",Y. [4] (b) Find E(T) and Var(T) (5] (c) If we assume that M = ... = Hn and a = ... = 0, what does the the successive geometric average, lim (II",Y), converge in probability to? Justify...
(5) Let Y,... Y2 be independent random variables from a distribution with distribution function P(У у-F(y), and density function f(s). Now let yl) be the minim um of all the observations. Show that the density function of Yu) is given by ow let Y(1 Yo ()n(1 - F(w)-f(w) Hint: First write out the CDF. P(W1) y), then using independence of the observations put it in terms of the distribution function F(), and then take the derivative to get the density.]
A random variable X is said to follow a lognormal
distribution if Y = log(X) follows a normal
distribution. The lognormal is sometimes used as a model for
heavy-tailed skewed distributions.
please answer the follow:
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Ex 2
Definition: A random variable X is said to have a binomial distribution and is referred to as a binomial random variable, if and only if its probability distribution is given by P(X-x)"C.pq" for x -0, 1,2,.., If X~B (n, p), then . E(X)= np and Var(X)=np(1-p) Notation for the above definition: n number of trials xnumber of success among n trials p probability of success in any one trial q probability of failure in any one trial Example...
Suppose that X is a continuous random variable with probability
distribution
Suppose that X is a continuous random variable with probability distribution O<x<6 18 (a) Find the probability distribution of the random variable Y-10X 3. fr o) 2 Edit for Sy s (b) Find the expected value of Y
A random variable Y is a function of random variable X, where y=x^3 and fx(x)=1 from 0 to 1 and =0 elsewhere. Determine fy(y). Ans: fy(y)=(1/3)y^(-2/3) for 0<y<1