3. In this question, you will identify the distribution of the sum of independent random variables. I expect you wi...
2. The joint density of X and Y is given by Say 0SX S1,0 Sy sa fxy(x,y) = {o otherwise. (a) Find fxiy (ay). (b) Set up the integrals (do not evaluate) for evaluating Cov(X,Y). 3. In this question, you will identify the distribution of the sum of independent random variables. I expect you will find that the mgf approach is your friend. (a) Let X and Y be independent Poisson random variables with means X, and A2, respectively, and...
Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is Y 1Xi Find/prove the mgf of Y and find E(Y), Var(Y), and P (8 Y 9) if a) X1,.,X4 are Poisson random variables with means 5, 1,4, and 2, respectively. b) [separately from part a)] X,., X4 are Geometric random variables with p 3/4. i=1
Let X1 and X2 be independent random variables with means μ1 and μ2, and variances σ21 and σ22, respectively. Find the correlation of X1 and X1 + X2. Note that: The covariance of random variables X; Y is dened by Cov(X; Y ) = E[(X - E(X))(Y - E(Y ))]. The correlation of X; Y is dened by Corr(X; Y ) =Cov(X; Y ) / √ Var(X)Var(Y )
2. SUPPLEMENTAL QUESTION 1 (a) Toss a fair coin so that with probability pheads occurs and with probability p tails occurs. Let X be the number of heads and Y be the number of tails. Prove X and Y are dependent (b) Now, toss the same coin n times, where n is a random integer with Poisson distribution: n~Poisson(A) Let X be the random variable counting the number of heads, Y the random variable counting the number of tails. Prove...
In this problem we show directly that the sum of independent Poisson random variables is Poisson. Let J and K be independent Poisson random variables with expected values α and respectively. Show that Ņ J+K is a Poisson random variable with expected value α+β Hint: Show that 72 Pk (m)P, (n-m and then simplify the summation be extracting the sum of a binomial PMF over al possible values. In this problem we show directly that the sum of independent Poisson...
3. (PMF – 8 points) Consider a sequence of independent trials of fair coin tossing. Let X denote a random variable that indicates the number of coin tosses you tried until you get heads for the first time and let y denote a random variable that indicates the number of coin tosses you tried until you get tails for the first time. For example, X = 1 and Y = 2 if you get heads on the first try and...
Problem 5. Indicator variables S points possible (graded) Consider a sequence of n 1 independent tosses of a biased coin, at times k = 0,1,2,...,n On each toss, the probability of Heads is p, and the probability of Tails is 1 -p {1,2,.., at time for E resulted in Tails and the toss at time - 1 resulted in A reward of one unit is given if the toss at time Heads. Otherwise, no reward is given at time Let...
A fair coin is tossed twice. Let X and Y be random variables such that: -X = 1 if the first toss is heads, and X = 0 otherwise. -Y = 1 if both tosses are heads, and Y = 0 otherwise. Determine whether or not X and Y are independent. So far, I have determined the the joint probability distribution as follows: x = 0 x = 1 y = 0 2/4 1/4 y = 1 0 1/4
3, Let X, X2,X, be independent random variables such that Xi~N(?) a. Find the distribution of Y= a1X1+azX2+ i.(Hint: The MGF of Xi is Mx, (t) et+(1/2)t) + anXn +b where a, 0 for at least one b. Assume = 2 =n= u and of- a= (X-)/(0/n) ? Explain. a. What is the distribution of The Sqve o tubat num c. What is the distribution of [(X-4)/(0/Vm? Explain.
Practice problems using various statistical methods If n independent random variables X have normal distributions with means μ and the standard deviations σ , then determine the distribution of a. I. X-E(X) var(X) C. 2. If n independent random variables Xi have normal distributions with means μί and the standard deviations σί, then determine the distribution of a. b. Y -a1X1 + a2X2+ + anXn (ai constant) X-E(X) Vvar(X) 3. What is CLT? Proof briefly? What are t-, Chi-squared- and...