(a)
b) This is nothing but a standard normal variate and as variance of X and Y are same. So, one can use .
c)
d)
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2. Suppose that X1, ..., Xd N(0,9) and suppose that Y1, ..., Y10 d (1,9). Assume...
Let X1,...,X10 be a random sample from N(θ1,1) distribution and let Y1,...,Y10 be an independent random sample from N(θ2,1) distribution. Let φ(X,Y ) = 1 if X < Y , −5 if X ≥ Y , and V= φ(Xi,Yj) . 1. Find v so that P[V>=v]=0.45 when 1=2. 2. Find the mean and variance of V when 1=2. 10 10 2 We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe...
Suppose that (X1, X2) N (0,0,1,1,0). It follows from this that the joint PDF of (X1, X2) is given by Ixvx:(21,2) = exp (1} (27 +23)) Furthermore, if 1 Y Tā (X1 + X2) and Y2 (X1 - X2) Then (Y1,Y) ~ N(0,0,1,1,0) as well. (a) If X1 <X2, what are the possible values of Y¡ and Y2? (b) If Y, <0, what are the possible values of Xi and X,? (c) What is the marginal distribution of Yg? (d)...
Pa Suppose that (X1, X2) ~ N(0,0,1,1,0). It follows from this that the joint PDF of (X1, Xy) is given by fxıx(31,09) - exp(+34 +23) Furthermore, if GE Thte and 12(x3 + x2) - tao V5 (Xi-X) que Then (Y1,Y,) ~ N(0,0, 1, 1,0) as well. (a) If X, < X2, what are the possible values of Y1 and Y2? (b) If Y2 < 0, what are the possible values of X, and X,? (c) What is the marginal distribution...
2. Suppose that X1, X2, ..., Xn " N(41,01) and Yı,Y2,...,Ym * N(H2;02) are two independent random samples. (a) What is E[X - Ÿ]? (b) Find a general expression for Var[X – Ý), and use this to find an expression for the standard error ox-ý = StDev(X – Ỹ). (c) Suppose that of = 2 and o = 2.5, and also that n = 10 and m = 15. Determine the probability P(|X – Ý - (µ1 – 42)| <...
1. Suppose we are going to sample n individuals and ask each sampled person whether they support policy A or not. Let Yi Y0 otherwise 1 if person i in the sample supports the policy, and (a) Assume Y1, , Yn are, conditional on θ. 1.1.d. binary random variables with expec- tation θ. Write down the joint distribution Pr(Yi-yi, . . . ,Ý,-yn(9) in a compact form. Also write down the form of Pr(> Ý,-y|0) (b) For the moment, suppose...
Question 2 Suppose you have a fair coin (a coin is considered fair if there is an equal probability of being heads or tails after a flip). In other words, each coin flip i follows an independent Bernoulli distribution X Ber(1/2). Define the random variable X, as: i if coin flip i results in heads 10 if coin flip i results in tails a. Suppose you flip the coin n = 10 times. Define the number of heads you observe...
Exercise 2, (a:3, b:4, c:4, d:3, e:3, f:3pt.) A compulsive gambler visits a casino and sees a row of n gambling machines ("one armed bandits") He cannot stop himself from playing on each machine until he has won. The i-th machine has probability pi E (0,1) of winning. (a) Let Xi be the number of times he play machine i. Give the pmf of X4 (b) Let M min(X1,... , Xn) be the least number of times he play the...
22) Find d-criticat values, and X1,h' c-0.90 and n-15. 4)5629 and 26.119 8) 6 571 and 23. D) 4.660 andd 29.133 )4.0% and 31.319 and X1, for c-099and n-10, 23) 23) Find the critical values, A) 2.088 and 21.666 B) 2 558 and 23.209 D) 2.156 and 25 188 C) 1.735 and 23.587 Assume the sample is taken from a normally distributed population and construct the indicated confidence interval. 24) Construct the indicated confidence intervals for the population variance o2...
4. Setup: Suppose you have observations X1,X2,X3,X4,X5 which are i.i.d. draws from a Gaussian distribution with unknown mean μ and unknown variance σ2. Given Facts: You are given the following: 15∑i=15Xi=0.90,15∑i=15X2i=1.31 Bookmark this page Setup: Suppose you have observations X1, X2, X3, X4, X5 which are i.i.d. draws from a Gaussian distribution with unknown mean u and unknown variance o? Given Facts: You are given the following: x=030, =1:1 Choose a test 1 point possible (graded, results hidden) To test...
Suppose that X1, X2, . . . , Xn is an iid sample of N (0, σ2 ) observations, where σ 2 > 0 is unknown. Consider testing H0 : σ 2 = σ 2 0 versus H1 : σ 2 6= σ 2 0 ; where σ 2 0 is known. (a) Derive a size α likelihood ratio test of H0 versus H1. Your rejection region should be written in terms of a sufficient statistic. (b) When the null...