Let X be a Poisson random variable with A - :3 and Y be a Poisson random variable with X = 4. Assume X and Y are independent. Find the following probabilities. (a) P(X <3, Y <3) (b) P(X>1,Y < 4)
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- Let X and Y be independent discrete random variables and suppose that X+Y=2. Show that X and Y are constant random variables.Let X₁, X₂ be 2 mutually independent discrete random variables. The first variable X₁ can be a whole number from 1 to 5. Its distribution function is 6 - i m₁ (i) 15 The second variable X₂ can be a whole number from 1 to 6. It has a uniform distribution. First, find the exact values (using fractions) of P(X₁ = 3) = 1/5 : P(X₂ = 5) = 1/6 P(X₁ = 3 and 3 and X₂ = 5) = 1/30 X₂ # Let Y denote the maximum of the X¿'s. Find the probability that Y = 1: P(Y = 1) = 1 Find the probability that Y = 2, P(Y= 2) = 2/45 Hint: Use cases based on what X₁ and X₂ are.let X(n,p) denotes binomial random variable with parameters n and p. Show that X(n+1,p) >st ( greater than equal ) X(n,p)
- Suppose X and Y are independent random variables with X - B (n, p) and Y ~ B (m, p). Show that X +Y ~ B (n + m, p).Q1:Show that: If X1, X2, , Xn are independent random variables and X = X1 + X2 ++Xn, then Q2: What is the expectation and the variance of RV X, where X represents the out come throwing a die? Q3: Find the expectation and the variance of X, where X is binomial random variable X - Binomial (n, p), Var(X)? 1 Chapter 3 Let X be a discrete random variable with range Rx = {1, 2, 3, ...}. Q4: Suppose the PMF of X is given by 1 for k = 1, 2, 3, ... 2k Px (k) = a) Find and plot the CDF of X, Fx (a). b) Find P(1 < X < 3). Chapter 3 Functions of Random Variables Q5: Example. Let Rx = {0, T T 37 , such that 4'2 4. Find E[sin(X)]. Q6: A machine produces a defected items with a probability of 0.1. What is the probability that in a sample of three item will have at most one item defected? Q7: For any independent X, and Y random variables, E[XY]=E[X]E[Y]. Show that? Q8: Suppose RV X has the following PMF: p(0)=0.2, p(1)=0.5, p(2)=0.3. Find E[X], E[X^2], E[X]^2. 3 Chapter 3Let X ~ N(1,2) and Y ~ N(4, 7) be independent random variables. Find the probability P(X + Y > 0):
- Suppose X is a random variable such that E(X) = 90 and Var(X) = 7. Compute thefollowing: E(4X + 12) Var(−X) SD(4X)E(X^2) E(X + 3)^2Let X(1), X(2), ..., X(n) be the order statistics of a set of n independent uniform (0, 1) random variables.Find the conditional distribution of X(n) given that X(1) = s(1), X(2) = s(2), ..., X(n-1)=s(n-1).Each of the random variables X and Y takes only 3 values {1,2,3} with the following probabilities: 1 1 0 1/6 1/6 y 2 1/6 0 1/6 3 1/6 1/6 0 2.