Let the mutually independent random variables X1, X2, and X3 be N(0, 1),
N(2, 4), and N(−1, 1), respectively. Compute the probability that exactly two of
these three variables are less than zero.
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Let the mutually independent random variables X1, X2, and X3 be N(0, 1), N(2, 4), and N(−1, 1), respectively. Compute the probability that exactly two of these three variables are less than zero.
Let X1, X2, and X3 be three independent, continuous random variables with the same distribution. Given X2 is smaller than X3, what is the conditional probability that X1 is smaller than X2?
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3). 3. (25 pts.) Let X1,...
Let X1, X2, X3 be independent random variables with E(X1) = 1, E(X2) = 2 and E(X3) = 3. Let Y = 3X1 − 2X2 + X3. Find E(Y ), Var(Y ) in the following examples. X1, X2, X3 are Poisson. [Recall that the variance of Poisson(λ) is λ.] X1, X2, X3 are normal, with respective variances σ12 = 1, σ2 = 3, σ32 = 5. Find P(0 ≤ Y ≤ 5). [Recall that any linear combination of independent normal...
Let the independent random variables X1 and X2 have binomial distribution with parameters n1 = 3, p =2/3, and n2=4, p=1/2 respectively. Compute P(X1 = X2).Hint: List the four mutually exclusive ways that X1 = X2 and compute the probability of each.
Let X1,X2 be two independent exponential random variables with λ=1, compute the P(X1+X2<t) using the joint density function. And let Z be gamma random variable with parameters (2,1). Compute the probability that P(Z < t). And what you can find by comparing P(X1+X2<t) and P(Z < t)? And compare P(X1+X2+X3<t) Xi iid (independent and identically distributed) ~Exp(1) and P(Z < t) Z~Gamma(3,1) (You don’t have to compute) (Hint: You can use the fact that Γ(2)=1, Γ(3)=2) Problem 2[10 points] Let...
14. Let X1, X2, X3 be independent random variables that represent lifetimes (in hours) of three key components of a device. Suppose their respective distributions are exponential with means 1000, 1500, and 2000. Let Y be the minimum of Xi, X2, X3 and compute P(Y 1000).
4.) Let X1, X2 and X3 be independent uniform random variables on [0,1]. Write Y = X1 + X, and Z X2 + X3 a.) Compute E[X, X,X3]. (5 points) b.) Compute Var(x1). (5 points) c.) Compute and draw a graph of the density function fy (15 points)
3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) = 2. o if 0<x< 1 else Find EY] where Y = min{X1, X2, X3, X4}.
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
If X1, X2, and X3 are three independent Uniform random variables (Xi-Unif(0,1)) a) Use the convolution integral to find density function of Z-x1+X2+X3. b) What is E[Z]? independent Uniform random variables (Xi-Unifo,1): If X1, X2, and X3 are three independent Uniform random variables (Xi-Unif(0,1)) a) Use the convolution integral to find density function of Z-x1+X2+X3. b) What is E[Z]? independent Uniform random variables (Xi-Unifo,1):