3.3.24. Let X1, X2 be two independent random variables having gamma distribu-tions with parameters a₁ = 3, B₁ = 3 and a2 = 5, B2 = 1, respectively.(a) Find the mgf of Y = 2X₁ +6X2.(b) What is the distribution of Y?

and denote independent random variables.Both have the gamma distribution with the following…

Answered: 3.3.24. Let X₁, X2 be two independent…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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1. The random variables §, §1, §2,... are independent and identically distributed with 1/4 and P(§ = j) = c/j for j = 1,2,3. Let X₁ = 0 and Xn = max(§₁, ..., Èn) for n = 1, 2,.... distribution P(§ = 0) = (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix. (d) Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). (e) Calculate P(Xn = 0). (f) Calculate P(X₁ = 3, X₂ = 1|X₁ = 3).
We have the following information about the random variables X and Y: |x = 1, |y = −1.5 , ox = 0.25, c} = 1. o Calculate the variance of Z = 3X + 5Y, a) when the coefficient of correlation is p(X, Y) = 0.46: 0²77 34.15 Z b) when X and Y are independent random variables: 0²2 27.75 X
Q. 4 Let X be a chi-square random variable with degree of freedom 6 (a) Find the quantile Q(0.95) and Q(0.050); (b) Let Y be a gamma random variable with a =3 and ß =- Then find 4 P(3.28< X < 25.2).
Let the following simple random sample X1, X2, · · , X11 following: 1. Binomial pmf. (11, ¾); 2. Uniform pmf; 3. Uniform pdf (0, a); 4. Exponential pdf with (µ). Find the corresponding pmf/pdf of Y1 , Y4, Y7 and F(Y;) where Yi < Y½ < .. < Y6 < ..< Yı1.
Q. Fourteen (V, X) : There are no correlated random variables X,Y with a? = 2a where Corr(X,Y) = a
Number...2
4.52 Random variables X and Y follow a joint distri- bution S2, 0
A sample of 20 ordered pairs (x,y) has a value for "r" of r = 0.485 Which of the following statements are true? At the 5% level of significance, a positive correlation exists At the 1% level of significance, a positive correlation exists At the 1% level of significance, a negative correlation exists At the 5% level of significance, a negative correlation exists
Q 6.1. Suppose Z = (Z₁, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., for i ≤ 3, Zi~ N(0, 1) are i.i.d. random variables. Each of the random variables, (a)–(d), on the left is equal in distribution to exactly one random variables, (1)–(4), on the right. Pair up according to "equal in distribution" and explain briefly your reasoning. X₁ (a) X₂ (b) (x₂) - (2) (c) (x²) - (¹/1² 1/√²) (2₁) (3/√2 = √2 = (V2) Z₁ (d) (x) - (1) (²) X2 Y₁ (9)-(-) (2) (1=1) Y₂ (1) Y₁ (²) (4) - (1/² (3) (1/√√2 1/√2-1/√2) (x₁) = (²₁ ² ✓/³²) (3) 2 2 √2 1 Z3 Y₁ → ()-()) (4) = Y₂ 1/2) (2) 1) (2)
Q 6.1. Suppose Z = (Z1, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., for i ≤ 3, Zi~ N(0, 1) are i.i.d. random variables. Each of the random variables, (a)-(d), on the left is equal in distribution to exactly one random variables, (1)-(4), on the right. Pair up according to "equal in distribution" and explain briefly your reasoning. (a) (X₂) (¹) (X) = (2) 1/√2 (Ⓒ) (x₂) - (3/1² ¹1/1²) (2) (c = 2 0 = 2 Z₁ 1) (²) 3 (4¹) (X) = (²¹) (1) (2) (3) Y₁ 1 (29) - ()) (2) = Y₂ 1 Y₁ (121) = (1/² √₂) (2) (1/√2 1/√2 /2 −1/√√2, Y₂ X₁ X₂ = 22 1 1 2 0 Y₁ 2 1 (4) (2)) = (²) (²) 1 1 Z₁ Z₂ Z3

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3.3.24. Let X₁, X2 be two independent random variables having gamma distribu- tions with parameters a₁ = 3, 3₁ = 3 and 02 = 5, 3₂= 1, respectively. (a) Find the mgf of Y = 2X₁ +6X₂. (b) What is the distribution of Y?