Question 8 random from an interval of length L. Find the probability that none of the...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4. 3 marks (b) Suppose X1 and X2 are two iid normal N(μ, σ*) variables. Define Are random variables V and W independent? Mathematically justify your answer 3 marks (c) Let C denote the unit circle...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4 3 marks (b) Suppose X, and X2 are two iid normal N(μ, σ2) variables. Define Are random variables V and W independent? Mathematically justify your answer. 3 marks] (c) Let C denote the unit circle...
3. [20 marks] Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and X3 have the joint probability function where x = (zi, 2 2:23), θ = (θί, θ2), n = x1 + 2 2 + x3, θι, θ2 > 0 and 1-0,-26, > 0. (a) [4 marks] Find the maximum likelihood estimator θ of θ. (b) [4 marks] Find that the Fisher information matrix I(0) (c) [4 marks] Show that θ is an MVUE. (d)...
Let X1, X2, and X3 be a random sample from a discrete distribution with probability function g(x) =x/10 for x= 1, 2, 3, 4 and g(x) = 0 otherwise. What is P(X1< X2< X3)?
3. [20 marks] Consider the multinomial distribution with 3 categories, where the random variables Xi, X2 and X3 have the joint probability function where x = (zi, 2 2:23), θ = (θί, θ2), n = x1 + 2 2 + x3, θι, θ2 > 0 and 1-0,-26, > 0. (a) [4 marks] Find the maximum likelihood estimator θ of θ. (b) [4 marks] Find that the Fisher information matrix I(0) (c) [4 marks] Show that θ is an MVUE. (d)...
4 0/6.66 points | Previous Answers LarCalcET7 5.3.075 Find the Riemann sum for f(x) x2 +3x over the interval [0, 8] see figure), where xo = 0, x1 = 1, x2-2, x3 = 6, and x,-8, and where C1-1, c2-2,C3-5, and C4-8. 302 10아 80 60 40 20 10 6 4 8 2 20
4 0/6.66 points | Previous Answers LarCalcET7 5.3.075 Find the Riemann sum for f(x) x2 +3x over the interval [0, 8] see figure), where xo =...
Let X1, X2, ..., Xn be a random sample from the distribution with probability density function f(x;t) = Botha, 0 < x < 2, t> -4. a. Find the method of moments estimator of t, t . Enter a formula below. Use * for multiplication, / for division and ^ for power. Use m1 for the sample mean X. For example, 7*n^2*m1/6 means 7n27/6. ſ = * Tries 0/10 b. Suppose n=5, and x1=0.36, X2=0.96, X3=1.16, X4=1.36, X5=1.96. Find the...
20 marksConsider the multinomial distribution with 3 categories, where the random variables X1,X2 and X have the joint probability function 123 [4 marks] Find the approximate distribution of Y = 2X1-X2, when the sample size n is large.
20 marksConsider the multinomial distribution with 3 categories, where the random variables X1,X2 and X have the joint probability function 123
[4 marks] Find the approximate distribution of Y = 2X1-X2, when the sample size n is large.
4. Let 8 >0. Let X, X2,..., X, be a random sample from the distribution with probability density function S(*;ð) - ma t?e-vor x>0, zero otherwise. Recall: W=vX has Gamma( a -6, 0-ta) distribution. Y=ZVX; = Z W; has a Gamma ( a =6n, = ta) distribution. i=1 E(Xk) - I( 2k+6) 120 ok k>-3. 42 S. A method of moments estimator of 8 is 42.n 8 = h) Suggest a confidence interval for 8 with (1 - 0) 100%...
a) An exponential random variable X has probability densityuntion: pdx]- forx20 Give a formula for the generating function of X b) A Beta random variable X has probability density function: pdfx] 6(1-x)x for0SxSI Give a formula for the generating function of X. (Tip: You can calculate: by first caleulatinge dx and differentiating with respect to t twice L.4) The generating function for the sum of independent random variables a) Given two independent random variables, X1 with generating function GX1[t] and...