6.5 Let X have the uniform distribution on [0,1]. That is, it has density f(x) = 1 for 0≤x≤ 1, and is otherwise 0. For 0 < p ≤ 1 find the quantile function that returns b, where P(X ≤ b) = p.
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![6.5 Let X have the uniform distribution on [0,1]. That is, it has density
f(x) = 1 for 0≤x≤ 1, and is otherwise 0. For 0 < p ≤ 1 find the quantile
function that returns b, where P(X ≤ b) = p.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdac418c7-7b26-43d7-bc46-18689988fbfb%2F8910141f-64a7-4111-9579-83556edd0f34%2Fw8c1yzt_processed.jpeg&w=3840&q=75)
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- Let X and Y be independent v.a with uniform distribution on the interval (0, 20) and (0, 30) respectively. Find the function f_X+Y (z).Let X1 and X, be independently and uniformly distributed over the interval (0, a). Find the p.d.f. of (a) U = X1 + X2. (b) W = X1 – X2.Q. 5 Let X be a continuous uniform random variable on [-a, a]. Find the pdf for Y = g(X), where 0, x b. g(x) = 3x,
- Let X has uniform distribution on [-1,1]. Find E (X|X²).Suppose that X is uniformly distributed on [0, 4]. Define Y = 2X + 5. Compute the pdf and cdf of Y.Let f, g be probability densities such supp(f) c supp(g). show yes i.i.d. X1,..., X, L(g) then f(X;) Wi g(X;) is hopeful 1. Argue why the weights need to be re-normalized even when the normalization constants of the densities f and g are known.
- Suppose that Y has a gamma distribution with parameters a and . a) If d is any positive or negative value such that a+d>0, show that ed r(a+d) E (Y*) – T(a) b) Use the result in part (a) to give an expression for E ( /1/Y) . What do you need to assume about a ? c) Show that, with d = 1, the result in part (a) gives E(Y) = a0Let X be the random variable having pdf f(x) = 1,0 < x < 1, zero e.w. Compute the probability of the range of the random sample of size 4 (X1, ., X4) is less than 1/2. Hint: Range is R = X(4) – X(1).Fill out the table giving the joint and marginal PMFs for X and Y. Find E[X] and E[Y]. Find the covariance of X and Y. Are X and Y independent?
- Let f, g be probability densities such supp(f) c supp(g). show yes X1, ... , Xn X1,..., X, iid. L(g) then f(X;) W; = g(X;) is hopeful 1. Argue why the weights need to be re-normalized even when the normalization constants of the densities f and g are known.27. If X is uniformly distributed over (0, 1) and Y is exponentially distributed with parameter λ = 1, find the distribution of (a) Z = X + Y and (b) Z = > X/Y. Assume independence.Suppose X is uniformly distributed over [0, 4] and Y is uniformly distributed over [0, 1]. Assume X and Y are independent. Let Z = X + Y. What is fz(4.5)?