**2.4.1.** Show that the random variables \( X_1 \) and \( X_2 \) with joint pdf\[f(x_1, x_2) = \begin{cases} 12x_1x_2(1-x_2) & 0 < x_1 < 1, \, 0 < x_2 < 1 \\ 0 & \text{elsewhere} \end{cases}\]are independent.**Explanation:**The joint probability density function (pdf) is given by \( f(x_1, x_2) \). The function equals \( 12x_1x_2(1-x_2) \) only when both \( x_1 \) and \( x_2 \) fall between 0 and 1. Everywhere else, the function is 0. To determine if \( X_1 \) and \( X_2 \) are independent, one must check if the joint pdf can be expressed as the product of the individual pdfs of \( X_1 \) and \( X_2 \).

Here Marginal pdf of X1 = f(x1) =f(x1)=∫01f(x1,x2) dx2 =∫0112x1x2(1-x2) dx2 =∫0112x1(x2-x22)dx2 =12x1(x222-x233)10 = 12x112(12-02)-13(13-03) =2x1 Here Marginal Pdf of X2 = f(x2)=∫01f(x1,x2) dx1 =∫0112x1x2(1-x2) dx1 =∫0112x2(x1-x1.x2)dx1 = ∫0112x2(1-x2).x1 dx1 = 12x2(1-x2)[x122]01 On solving, f(x2)=6x2(1-x2) Now ,f(x1).f(x2) = f(x1,x2) [ Because 2x1(6x2(1-x2))=12x1x2(1-x2) = f(x1,x2)] Hence, to conclude Random Variables X1 and X2 are independent.

Answered: 2.4.1. Show that the random variables…

Math

Advanced Math

2.4.1. Show that the random variables X1 and X2 with joint pdf f(21, #2) = { 0 1212(1- 2) 0

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

4. Suppose X1,... , Xn is a random sample of size n drawn from a Poisson pdf where A is an unknown parameter. Show X X is an unbiased estimator for X
4. Two independent random variables x₁ and x₂ are both uniformly y₁=X₁X2/(x₁+x2). Hint: distributed between 9 and 11. Find the pdf of introduce the dummy variable y2=X2 and use the approach of two functions of two variables.
9.1.4 X1, X2 and X, are iid continuous uniform random variables. Random variable Y = X, + X2 + X, has expected value E[Y] = 0 and variance of = 4. What is the PDF fx,(x) of X,?
3. Consider two random variables X₁ and X2 whose joint pdf is given by for x₁0, x2 > 0, x₁ + x2 < 2, { Find the pdf of Y = X₂ - X₁. f(x1, x₂) = NIT otherwise.
7.2.3. X is the continuous uniform (-5, 5) random variable. Given the event B = {|X| < 3}, find the (a) conditional PDF, fx|B(x), (b) conditional expected value, E[X|B], (c) conditional variance, Var[X|B].
7. A discrete random variable X taking non-negative integer values has probability gen- erating function: 1 rx(t) = (2- t)2 (a) Calculate P(X = 2).
Part II: Sections 2.1 2.7 7. Suppose that X is a discrete random variable with pmf f(x) =D c.14-피 . 뉴- 21 for r=-3, 3, 6, where c is a constant. (a) Find c. Compute E ( ~LX +) (b) 3.
der a random sample X1, X2, ..., X, having the pdf, f(r; 0) 1
Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?
4.2 The probability distribution of the discrete random variable X is f(x) = 3 (*)*** Find the mean of X. 3-x (¹)ª (²³)³—ª, x = 0, 1, 2, 3.
Bapu
Suppose there are three possible states of nature, and the class-conditional PDFS are the Cauchy distributions 1 p(rw;) = 1 i = 1,2,3. (1) 2 1+ bị Let ai = -2, a2 = 0, az = 2, bị = b2 = 1, b3 = 0.5. Let P(wi) = P(w2) = 0.4, P(w3) = 0.2. (a) Find the decision boundaries and the decision regions for the Bayesian decision (b) Calculate the probability of error for the classification done according to this