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- Example 24: Let X is a normal random variable denoted by N (0, ơ). Let Y =X°. Find the expected value of Y.Practice Question 2: Let X₁ a random variable with expected value (mean) E(X₁) = 17 and variance V(X₁) = 12. Let X₂ another random variable with expected value (mean) E(X₂) = 22 and variance V(X₂) = 16. The covariance between X₁ and X₂ is Cov(X1,X 2)=9. Let a new random variable Y defined as Y = 9X₁ + 10x₂. a. d. Explain briefly the meaning of the correlation of these two random variables X₁ and X2. e. Explain briefly if X₁ and X₂ are linearly independent not.4. Suppose that the random variable X has p.d.f. f(x) = { 1- |x – 1|, if 0 0.5. (c) Find the mean and variance of the random variable X.
- 4. The joint probability distribution of random variables, X and Y, is shown in the following table: X 1 0.48 0.02 Y 0.05 0.03 3 0.2 0.1 4 0.04 0.08 (a) Calculate P(X = 1|Y = 3). %3D (b) Find the mode of the joint probability distribution. (c) Calculate the variance V(Y|X = 5). (d) Calculate the variance V(3Y- X).If X; and independent, identically distributed (iid) Gaussian random variables with mean, µ, and variance, o², for i € {1, ..., n}, what is the distribution of the sample mean? How is the sample variance related to a common distribution? Sketch the distributions from problem 3 and label key points.cORLEM 2. Arandom variable X has the following probability Function find the value of K i) ii) Mean iii) Variance iv) P (X>3) v) p(112. i. State Central limit theorem and tell what is the main theme of it. ii. Show that if the random variables (X, Y) follow the probability density of a bivariate normal distribution and if the correlation p = 0, X and Y are independent. (Show that f(x, y) = g(x) h(y))7. If a random variable has the probability density function -1≤x≤3 otherwise f(x)=k(x²-1) =02. The random variable X is the amount of fructose in apples (in ounces) grown at a local orchard and is represented by the following cdf: 2(x +), 1Please help me with the solution step by step.1. The number of cellphones sold per day at a local cellphone shop, along with its corresponding probabilities, is shown in the table. Find the mean, variance and standard deviation of the probability distribution P(X) x P(x) x2 P(x) 1 0.15 2 0.35 3 0.40 4. 0.10 u = =x2 • P(x) = 2. Find the mean, variance and standard deviation of the probability distribution of the random variable X, which can take only the values of 1, 2, 3 and 4, given that P(1) = P(2) = 5 P(3) =; and P(4) = 5 %3D %3D %3D P(X) x P(x) x2 P(x) u = u = Ex2 • P(x) =Let X be a random variable representing number of times the system needs to be repaired before the claim of full failure. The probability mass function of X is given in the following distribution table: X1 2 3 1 3 1 f(x) 10 5 10 Then Variance of X is equal to: 0.84 1.01 1 O None of theseRecommended textbooks for youCalculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage Learning
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