If f and g are density
a. Show that
b. Use Jensen’s inequality and the Identity
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A First Course In Probability, Global Edition
- Let X be a continuous random variable with a density functionfX (x) = (c sin (πx) 0 <x <1 = 0 otherwise. Let Y = √X and determine the density function fY.arrow_forwardIf X and Y are jointly continuous with joint density function fx,y(x, y), show that X +Y is continuous with density function fx+Y(t) = | fx,Y (x,t – x)drarrow_forwardSuppose fz(z) is a function given by So(23 – 2) if -z<0 | fz(2) = otherwise for a fixed constant c. Can fz(z) be a density function? Explain.arrow_forward
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- For some 0 € (0, 1), define the density on the interval (0, 2) : f(x|0) = { 1 0x € (0,1], 0x = (1, 2). (a) Verify that f(x0) is a probability density function and draw a picture of it for 31.arrow_forwardLet the density function of a random variable X be given by f:(x)= 0x, 0SXS1 Compute the sufficient estimator of e log (X.) a) b) c) log (X.) -log (X) d) Answerarrow_forwardThe life time, in years, of some electronic component is a continuous variable with the density f(x) = ) = - (a) Find k. (b) Find cdf F(X). x/x7, 0, x ≥ 1 x < 1 OCarrow_forward
- Let F(v)=P(Zarrow_forwardFor some 0 € (0, 1), define the density on the interval (0, 2) : Vx € (0, 1], 0 0 Vx € (1,2). f (x|0) = { ₁- 1 - (a) Verify that f(x0) is a probability density function and draw a picture of it for 0 = ³1. Suppose X₁,. ,..., Xn is a random sample from f(x|0). (b) Compute the method of moments estimate for 0. (c) Is the estimator in (b) unbiased? Justify your answer.arrow_forwardLet X and Y be jointly continuous with joint probability density function f(x, y) and marginal densities fX(x) and fY(y). Suppose that f(x, y) = g(x)h(y) where g(x) is a function of x alone, h(y) is a function of y alone, and both g(x) and h(y) are nonnegative. a) Show that there exists a positive constant c such that fX(x) = cg(x) and fY(y) = (1/c)h(y). b) Use part (a) to show that X and Y are independent.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON