(6) The probability density function of the random variable X whose momentgenerating function () (1 + e¹) is:(a) , x = 0,1,2(c) ,x=0,1,2(b) , x = 0,1x+1(d) x¹, x = 0,13

Given Moment generating function=Mx(t)=1/2(1+et)

Answered: (6) The probability density function of…

A First Course in Probability (10th Edition)

10th Edition

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Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

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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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Q2 Let (X₁, X₂) be jointly continuous with joint probability density function = { -(x₁+x2), f(x₁, x₂) = x1 > 0, x₂ > 0 otherwise. Q2(i.) Sketch(Shade) the support of (X1, X₂). Q2 (ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂. Q2(iii.) Let Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and then find the probability density function of Y₁, i.e., fy, (y). Q2(iv.) Let Mx, (t) = Mx₂ (t) = (¹, for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. 1 - Q2(v.) Let Y₂ = X₁ – X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y₂, and using the moment generating function of Y₂, find E[Y₂]. Q2(vi.) Using the bivariate transformation method, find the joint…
Example of (a) , (b) is (12),(24),(1),(6) Example of (c) (i) or (ii) Thanks
A random variable has probability density function f(x) = 30x (1 - x) <(0 < xs 1). Compute its cumulative distribution F(x). O 10x3- 15x4 + 6x5 + 1 O 30x(1 - x) O 30x2 - 60x3 + 30x4 O 10x3 - 15x4 + 6x5 O none of these A Moving to the next question prevents changes to this answer. to search 63°F S End F10 Home %23 PrtScn F8 F2 F3 F4 F5 F6 F7 F9 近
5
a) Let Y be a random variable whose probability density function and cumulative distribution function is given by b) and, f(y;0) = ey-8 (1+ey-8)² F(y; 0) = 1- " 1 1+ey-8, -∞ 1. ii) Find the power of the test which rejects Ho if Y > 1. iii) If a test that rejects Ho given Y₁ ≥c is considered. What will be the value of c when the level of significance of the test is 0.1. The daily salaries (in rands) for an international bank are normally distributed with mean and standard deviation of 14.20 and 3.50, respectively. Suppose that the bank randomly selected fourty (40) tellers and hire them, paying each teller on average of 12.20 per day. Test at 1% level of significance whether the bank can be accused of paying poor salaries or not?
2) The probability density function of a random variable X is giyen by: [kx(2-x),03) 3)
Example 18.2 For the probability density of a system of random variables (X, Y): f(x, y) = 0.5 sin (x + y) (0 < x < 7,0 << 7), 2' determine (a) the distribution function of the system, (b) the expectations of X and Y, (c) the covariance matrix.
1. Verify whether the following functions can be considered as probability mass functions: (i) P(x = x) = x² + 1 18 (iii) P(X= x) = -, x = 0, 1, 2, 3 (ii) P(X - x) - ²-2, -, x= 1, 2, 3 2x + 1 18 -, x = 0, 1, 2, 3 [Ans.: Yes) [Ans.: No] [Ans.: No]
3 Let X be a random variable with probability Law P (X = r) = q, K for r = 1,2,3.. . 00, Find the moment generating function and mean of ihe random variable X. Let K+ q = 1.

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(6) The probability density function of the random variable X whose moment generating function () (1 + e¹) is: (a) , x = 0,1,2 (c), x = 0,1,2 (b) ,x=0,1 x+1 (d) x¹, x = 0,1