2 a) as expectation is linear.
.
2 b)
(Since is a one dimensional random variable, )
(Since expectation is linear)
Let X be a random variable with probability density function 2 (r > 1 0 otherwise. (a) Compute F)-P(X ) (the cumulative distribution function) for 1. Note that F(x) 0 for 1 (b) Let u-F(z). Invert F(-) to obtain 2 marks [1 mark 3 marks) F-1 (u), (z as a function of Your function should have:- Input: n - Number of samples to be generated. . Output: x - (xi, x2,, n) A vector x of n values from the...
1. (10 marks) random variable with density r(x). Let g: R - (a) Let X R be a (differentiable) function and let Y = g(X). Write expressions for the following ((ii)-(iv) should be in terms of the density of X (i) The integral f()d (ii) The mean E(X) (ii The probability P(X e (a, b) (iv) The mean E(g(X)) R be a smooth (1 mark (1 mark) (1 mark (1 mark) (b) Let z E R be a constant and...
Let X variable Y by be a normal random variable with mean 0 and variance 1. We define the random y2 if x 20, Y= (a For t E R, compute Mr()-Elen'], the moment generating function of Y. Compute EY
2. Let U be a continuous random variable with the following probability density function: 1+t g(t) = 1-t -1 < t < 0 0 <t<1 otherwise 0 a. Verify that g(t) is indeed a probability density function. [5] b. Compute the expected value, E(U), and variance, V(U), of U. (10)
Please explain Let Z N(0,1), and let X = max(Z, 0) 1. Find Fx in terms of Φ(t). Ís X a continuous random variable ? 2. Compute p(X0) 3. Compute E(X) . Find the PDF fxa(u) 5. Compute V(X) (Hint: use fxa found above Let Z N(0,1), and let X = max(Z, 0) 1. Find Fx in terms of Φ(t). Ís X a continuous random variable ? 2. Compute p(X0) 3. Compute E(X) . Find the PDF fxa(u) 5. Compute...
2. Let U be a continuous random variable with the following probability density function: 1+1 -1 <t<o g(t) = { 1-1 03151 0 otherwise a. Verify that g(t) is indeed a probability density function. [5] b. Compute the expected value, E(U), and variance, V(U), of U. (10)
2. Let U C R2 be simply connected and let to E U. Let g: U(oR2 be irrotational and of class C1. Assume that there exists r >0 such that B(zo, r) C U and g=0. Let γ be a closed sinile polygonal arc with range in U \ {zo), let「be its range, and let V be the bounded connected component of R2 \ Г. (a) Assume that V C U \ [xo) and prove that g=0. (b) Assume that...
2. Let U be a continuous random variable with the following probability density function: g(t) = 1+t -1 <t < 0 1-t 0<t<1 0 otherwise a. Verify that g(t) is indeed a probability density function. [5] b. Compute the expected value, E(U), and variance, V(U), of U. (10)
1. Find Fx in terms of φ (t). Is X a continuous random variable ? 2. Compute p(X 0) 3. Compute E(X). Hint: use the CDF expectation formula, and integration by parts. You may assume that lim, t"o(-t) 0 for all n 2 0. 4. Find the CDF Fx (u) 5. Compute V(X). Hint: use Fxa, and follow the same hint of part (3) 1. Find Fx in terms of φ (t). Is X a continuous random variable ? 2....
Let U be a continuous random variable with the following probability density function: g(t) = 1+t -1<t< 0 1-t 0<t<1 0 otherwise a. Verify that g(t) is indeed a probability density function. [5] b. Compute the expected value, E(U), and variance, V(U), of U. (10)