Example 1: X is uniformly distributed over |-1.1]. Calculate P(X1 2a) where a 0 using the...
Let X be an exponentially distributed random variable with parameter value 2. a) Use the markov inequality to estimate P(X >= 3). b) Use the chebyshev inequaity to estimate P(X >= 3). c) Calculate P(X >= 3) exactly.
Рroblem 5. Let X1. X2,.. be independent random variables that are uniformly distributed over [-1.1. Show that the sequence Yı , V2.... converges in probability to some limit, and ident ify the limit, for each of the following cases: (а) Ү, Хn/п. n
5. Let X be uniformly distributed in [0, 1]. Given X = x, the r.v. Y is uniformly distributed in 0, x for 0
Let X be uniformly distributed on [0,01, where θ (0,00) is an unknown parameter. (a) Given an 1.1.d. sample of size n, X1, .. . , Xn, construct an unbiased estimator of θ (b) Consider a specific type of decision rules d,(X) = cx, c 〉 0, and assume the quadratic loss function. For many values of c, c2 0, de is inadmissible. Comparing all de(X), specify value(s) of c which may make de (X) to be admissible.
0/1 point (graded) Let X be distributed uniformly over {-1 { Xw.p. 3/4, where w.p. means "with probability". Find Cov (X, Y) , 1f, and let Y- X w.p. 1/4., 2 Submit You have used 1 of 4 attempts Reset
Suppose that Y=cos(X), where X is uniformly distributed over the interval [0, 2Pi]. Determine the pdf of the random variable Y.
(a) Let X(t) sin t, where is uniformly distributed over the interval [0, 27]. Verify (i) The discrete-time process (t), t = 1,2,... is weakly, but not strongly stationary (ii) The continuous-time stochastic process X(t), t> 0 is neither weakly nor strongly stationary (a) Let X(t) sin t, where is uniformly distributed over the interval [0, 27]. Verify (i) The discrete-time process (t), t = 1,2,... is weakly, but not strongly stationary (ii) The continuous-time stochastic process X(t), t> 0...
13. Let Xand Xbe independently and uniformly distributed over the interval (0,a). Find the p.d.f. of (a) U = X1 + X2 (b) W = X1 - X,
5. Let X be uniformly distributed in [0, 1]. Given X = x, the r.v. Y is uniformly distributed in 0, x for 0<x<1 (a) Specify the joint pdf fxy(x,y) and sketch its region of support Ω XY. (b) Determine fxly(x1025). (c) Determine the probability P(X〈2Y). (d) Determine the probability P(X +Y 1)
If X is uniformly distributed over (0, 2), find the density function of Y = e X. The density can be given only on the interval (1, e 2 ) where it is non-zero.