無年度 - 主題課程_理工學院機率論：多元隨機變數#107950

十五. Consider the famous hat problemn. Suppose that n pcople throw their hats in a box and n each picks one hat at random. (Each hat can be picked by only one person, and cach assi of hats to persons is equally likely) Let X be a random variable that ropresents the number of people that get back their own hat. Which of the following statemnents is/are true? (A) When n =3, P(X =3)= (B) When n = 3, P(X = 1)= (C) When n = 4, P(X = 3) = 0. (D) When n = 4, E[X] = 2. (E) When n = 5, E[X] = 1.

十六. Consider the famous hat problem as in the previous problem. Let X, be a random variable that takes value 1 if the ith person selects his/her own hat, and takes value O otherwise. Let cov(Y, Z) = [Y . Z] -E[Y] ㆍE[Z] be the covariance of two random variables Y and 2. Which of the following statements is/are true? (A) When n = 4, cov(X₄, X₄) is .(B)When n = 4,cov(X₁,X₂)=(C) When n = 4, va=2. (D) When n = 4, Xi and Xz are statistically independent. (E) For any two random variables Y and Z, cov(Y, Z) ≥0.

X and Y are independent continuous random variables with probability density functions (PDFs) shown below. What is the probability of P(-1

(A) 1/4 (B) 2/3 (C) 3/4 (D) 5/8 (E) none of the above.

15. (5%) The random variables X₁, X₂,X₃ have the joint Gaussian pdf: What is the marginal Goint) pdf of Xi and X3? (A) 0. (B) (C) (D)(E) None of the above. 

16. (5%) According to Problem 15 above, for the 3✖3 covariance matrix of (X₁.X₂,X₃), determine the correct element values Ki of the covariance matrix. (A) (B) (C)(D) (E) None ofthe above.

(5%) Let X and Y have the joint probability density function f(x,y)=2,  0≤x≤y≤1. Which of the following is correct? (A) E[X] = 1 (B) E[Y] = 1/2 (C) E[Y²] =1/2 (D)Pr(o sX sY,O≤Y≤=1/2 (E) None of these

(二)求條件機率(conditional probability)(0|0)= P(X=0|Y=0)(5分)

Problem 7. (2.5 %) Consider two random variables X and Y, each taking val- ues in {1,2,3}. Let their joint PMF be such that for any 1 ≤ x, y  ≤ 3,

Then, 

(i) X and Y can be independent or dependent depending upon the strictly positive values
(ii) X and Y are always independent
(iii) X and Y can never be independent

(b) (4%) Find(x, y).

(c) (4%) Find.

(d) (5%) Find E[Y|A].

(b) (4%) Find the conditional probability density function .

(d) (8%) Let Z = 3X + 2Y. Derive the moment generating function . You have to give the derivation, not just the an- swer.

8.(10%) The joint p.d.f. of x and y is for some k. Deter-

mine the conditional p.d.f.. 

a)Derive the conditional cumulative distribution function.(5%)

c) Is the correlation coefficient = 0? Explain why. (5%)

(b) (5%) Find the conditional pdf

(c) (5%) Find the conditional pdf

(d) (5%) Find the conditional mean E[Y|X = x], for x≥ 0

(e) (5%) Find the conditional variance Var(Y|X = x), for x ≥ 0 

7.(8%) The random variable x is uniform in the interval [-5, 5]. Define y = Find C.D.F. F_y(y) and p.d.f. f_y(y). 

(10%) Suppose that X is a continuous random variable with cumulative distribution function (CDF) Fx(x) and probability density function (pdt) fx(x), respectively. Let Y = |2X|. Find the CDF of Y.