με6. Let X and Y be jointly normal random variables with parameters x = = 0²3, rx = 16, Y =25, and p = 3/5, i.e. (X,Y) ~ N₂(3, 1, 16, 25, 3/5). Find1, 0²=(a) P(3

Answered: 6. Let X and Y be jointly normal random…

MATLAB: An Introduction with Applications

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Author:Amos Gilat

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3. Let X1, X2, ..., Xn be a random sample from the population N(µz,0²) and Y1, Y2,...,Ym a random sample from the population N(µy, o2), where both means (z and Hy) are assumed known. Note that the two distributions have a common variance of o². (a) If the X's and Y's are independent, show that the MLE for the common variance is: E (Xi - H2)² +E(Y; - Hy)? n+ m (b) Is the MLE found above unbiased for o2?
18. Assume that Z, V, and V, are independent random variables with Z ~ N(0, 1), V ~x(5), and V ~ x²(9). Find the following: (a) P[V + V½ 0.611/V]. (d) P[V/V, < 1.450]. (e) The value b such that P < b + V2 = 0.90.
Let X1, X2,.., X, be independent identically distributed random variables with each X, having a probability mas function given byP(X; = 0) = 1-p P(X; = 1) = p, where Osps1. 1 EXj, then E(Y)= j = 1 Define the random variable Y = n Select one: а. 1 b. p C. 0 d.
(a) Let X, Y be independent random variables with respective PMFS Px(x) = (-) )* > Py (Y) = (-) )", y=1, 2, 3, . . . .2 , х-0, 1, 2, 3, . . Find P(X+Y=3)
Suppose the joint DF of the random variables X and Y is given by: (0. x0, and evaluate the probabilities at these points.
Q2/(a) Let Xbe a random variable with p.d.f. 1
4. Continuous random variables X and Y are independent, with parameters ux = 30, ly = 40, Ox = 15, and ơy = 5. Which of the following gives the correct values for ux+Y and ox+y? = 70 and ox+y = 14.1 (A) Hx+Y (B) Мx+ү %3D 70 and o x+y %3D 15.8 (C) Hx+Y = 70 and ox+y = 20 (D) Ux+Y = -10 and ox+Y =14.1 (E) Ux+y =-10 and ox+y = 15.8
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
Let X1, X2, ..., X50 be 50 independent and identically distributed exponential random variables with expected value 2. (i) Compute P (X; 3)) (iv) Let Y = 3e²X: + 4X¡. Find E(Y).
Suppose that the random variables Y1,..., Y(n > 2) satisfy Y - Ba, + Ei, i= 1,2,..., T, where 1,..., En are iid N(0, o), and both B and o are unknown. (a) Assume z1,..., In are fixed known constants. Here we observe Y1 = n,., Y observed data Y = y = (y1,..., Yn). Find a two-dimensional sufficient statistic of Y = (Yı,., Yn) for (8,0?). (b) Assume now that a1,...,n are random variables with a known joint distribution m(r1,..., In), and the r,'s are independent of G's (it is traditional in the linear regression to use lower case for independent variables r;'s). In this case, the observed data (Y, x) = {(Yi, r;)}i-1,n. Find a three- dimensional sufficient statistic of (Y, x) for (B,02). Im, and the

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6. Let X and Y be jointly normal random variables with parameters x = 3, o = 16, μy 1, o = 25, and p = 3/5, i.e. (X,Y) ~ N₂(3, 1, 16, 25, 3/5). Find (a) P(3