You wish to test whether a coin lands "heads" and "tails" with the same probability. Use p to denote the probability that the coin lands "heads".

(a) State the relevant null and alternate hypotheses.

(b) Suppose you choose to reject the null hypothesis when in 10 tosses you get more than 7 or less than 3 "heads".

i. What is the power of your test when the true probability of "heads" is 0.6?

ii. What is the probability that your test results in a Type II error when the true probability of "heads" is 0.3?

iii. Write an expression for the power function ?(p) of the test.

iv. What is the size of the test?

For each of the following transition matrices, determine whether the Markov chain with that transition matrix is regular: (1) Is the Markov chain whose transition matrix whose transition matrix is 0 0.5 0.5 0.5 0 0.5 0 0 regular? (Yes or No) (2) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.3 0 0.7 0 0 regular? (Yes or No) (3) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.6 0 0.4 1 0 0 regular? (Yes or No) (4) is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0 0 0.6 0 0.4 regular? (Yes or No) (5) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.3 0.2 0.5 0 1 0Problem 3. Consider the Markov chain shown in Figure 2. Figure 2: Problem 3 Markov chain 1' Let the initial distribution be Pr(A) = Pr(B) = 0.5. What is the probability distribution after one step? 2. What is the stationary distribution of the Markov chain? Q. 6 In this problem we deal with Bivariate Gaussians. Please read the lecture notes available under the 'Bivariate Gaussian RVs' section on Canvas (L16.pdf). The lecture notes summarizes the following facts about bivariate Gaussians: X and Y are bivariate Gaussian if their joint PDF is given by HX - HX fx,y (x, y ) = 2noxGY (1 - Pxy ) 1/2 exp 2(1 -PxY) - 2PxY Ox Ox This PDF is specified by 5 parameters which correspond to the marginal means, variances and correlation, i.e., (X, Y) ~ N(ux, MY, ox, 67, pxY). Some interesting facts about bivariate Gaussians are listed below: . Marginals are Gaussian: X ~ N(ux, Gx ) and X ~ N(ux, 63). . Bivariate Gaussian RVs are uncorrelated iff they are independent: If pyy = 0 then the joint density can be written as fx, y (x,y) = fx(x) fr ())= 2nox -e- ( x- ux )/20% 1 - ()-MY)2 /20% V2noy thus the RVs are independent. . A linear transformation of bivariate Gaussian is also bivariate Gaussian, but with new parameters. . Bivariate Gaussian and conditioning: For bivariate Gaussian RVs (X, Y) the conditional distributions fry and frix are also Gaussian, and are given by, frix (y | x) = fxr (x, y) - exp (y - HYLx (x))2) fx (x) V2noylx 20 yx ~ N( ur x, 0; x ), where E[Y [ X] = Myx (X) = Hy + Pxy ox (X -ux ) and Gyx = of (1 - Pxy). Suppose that X ~ N(0, 1) and Y ~ N(1, 2) are two independent Gaussian random variables. Define random variables U and V as U = X + Y, V = X + ay for some a. 1. What is the joint distribution of U, V in terms of o? 2. Find o (if it exists) such that U and V be independent. 3. Find the conditional distribution fulv (u|5)