10. The density function for a random variable X is given byJ / x + k(a) Find k.(c) Find u.f(x) =if 2 ≤ x ≤4otherwise(b)Find P(X> 2.5).(d) Find P(2 < X < μ).

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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7. Let X and Y be continuous random variables with joint probability density function given by fxy(xy): Sc, (0, 0 < x < y < 1 otherwise a) Find the value of c. b) Find the marginal distributions of X and Y. c) Find the conditional distribution of Y given X. d) Find the E(Y) and Var(Y).
4. Let x be a continuous random variable with the following probability density function 3 (2 – x) 0 < x< 2 f (x) = else (a) Find E(x) (b) Find Var(x)
8)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the joint probability density function of the order statistics X(2), X(4), X(6) for n = 7.
b) Let X₁, X2, X3,...,Xn be a random sample of n from population X distributed with the following probability density function: f(x;0)=√√2n0 0, -20₁ if -∞0 < x <∞0 otherwise (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. (iii) Check whether or not the estimator obtained in (ii) is unbiased. (iv) Find the Fisher information in this sample of size n about the parameter 0.
b) Let X₁, X2, X3.....Xn be a random sample of n from population X distributed with the following probability density function: ze zo, f(x;0)=√2m0 0, (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. if -∞
7. Let X and Y denote two continuous random variables. Let f(x,y) denote the joint probability density function and fx(x) and fy (y) the marginal probability density functions for X and Y, respectively. Finally let Z = aX + bY, where a and b are non-zero real numbers. (e) Derive an expression for Cov(Z) as a function of Var (X), Var(Y) and Cov(X,Y). [You may use standard results relating to variance and covariance without proof, but these should be clearly stated.]
Let X be the proportion of new restaurants in a given year that make a profit during their first year of operation, and suppose that the density function for X is ƒ(x) = 20x³(1 − x) Find the expected value and variance for this random variable. E(X) = = Var(X) 0 ≤ x ≤ 1
If X, and X, are two random variables having joint density function 2. f(x,,x, ) = +8x,x,* ,0Sx, S 2,0s x,sI 8. Find i. P(X, I) P(X, <1/ X, <0.5) P(X,+ X, <1) ii. iii. iv.
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10. The density function for a random variable X is given by f(x) J/x+k if2 ≤x≤4 ) = {{ x + k otherwise (a) Find k. (c) Find μ. (b) Find P(X> 2.5). (d) Find P(2 < X < μ).