3. Let X and Y be two random variables with and fy(y) = 10, fxy(x|y) = 2/3, if 0 ≤ y ≤1 otherwise, ye-, if r 20 0. otherwise.
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![3. Let X and Y be two random variables with
and
fy(y):
=
fxy(x|y) =
y²/3, if 0 ≤ y ≤1
0,
otherwise,
Find E[X].
Give your answers to two decimal places.
Sye-v², if x ≥ 0
0,
otherwise.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e9ca648-396a-488a-88dd-91544590f068%2F557ee7ba-c91a-4590-83f7-246de4cd9482%2F7a8bvoo_processed.jpeg&w=3840&q=75)
Given information:
Consider,
Step by step
Solved in 2 steps
- 4. Suppose we have random variables W and Q, and we know Var(W) = 25, Var(Q) = 10, and Cov(W, Q) = 2. Now, let A = 2W + 4Q. What is Var(A)?IV) When a current X amperes flows through a resistance Y ohms, the power generated is given by w=x²y watts. Suppose the current and resistance are independent random variables with densities b = f(x)=6x (1-x), 0≤x≤1 X If the CDF of Wis Fw (w) = wª +6w - 8w 3/2, b ≤w ≤c, where a, b and care constants then a = C= fy(y)=2y, 0≤x≤13. Let X be the random variable that takes on the integers {0, 1, 2, ..., 15} with equal probabilities. Define a new random variable Y = X + A, where A is a random variable that takes on the values {-1, 0, 1} with equal probabilities. If the RVs X and A are independent, find the mutual information between X and Y.
- 20Let JO, J1,..., J4 independent random variables according to the Ber (r;) law, where i = 0, 1,..., 4, respectively. We define the random variables Xi = min {JO + Ji, 1}, for i = 1, 2, 3, (a) Find the law of Xi , for each i = 1, 2, 3, 4. (b) Find the law of (X1, X2, X3, X4).9. Given that f(x, y) = (2x+2y)/2k if x = 0,1 and y = 1,4, is a joint probability distribution function for the random variables X and Y. Find: (f(x|y = 1)
- Let Z be a random variable with E(Z) = 12 and Var(Z) = 5. Based on the statement above, determine which of the following statements are true and false. Show complete solution. a) E(3Z + 10) = 46 b) E(Z^2) = 160 c) Var(10)=109. If X and Y are two random variables and let g(X) be a random variable. Show that (a) E[g(X) X=x] = g(x). (b) E[g(x)Y|X=x] = g(x) E[Y|X=x]. Assume that E[g(x)] and E[Y] exist.B4. Let X₁,... Xn ~ N(μ, o2) be independent random variables. 2 (a) From lectures we know (X=X) ²³. X₂ (b) Let n i=1 ~ x²(v). What is the value of v? n 1 *³ =, ²-₁, [(x₁ - x) ². Σ(x s n 1 i=1 Determine Var(s), that is, the variance of the sample variance. (c) Assume now that we have observed data ₁,...,n ER with sample variance s². Use the result from part (a) to find numbers an and b, such that [ans, bns] is an exact 95%-confidence interval for o².
