PLEASE HELP ME I CONFUSED
Posted: Fri Dec 24, 2021 10:09 am
PLEASE HELP ME I CONFUSED
, 2 Consider (X1,Y1, Z1),... (Xn, Yn, Zn), where the (Xi, Yi, Zi) are i.i.d. as random vectors. (For fixed i, the scalar random variables Xi, Yi, and Zį are not necessarily either independent or indentically distributed!) Each of the random variables Xi, Yi, and Zi takes values in {0, 1}. Suppose that conditioned on Zi, the random variables X, and Yį are independent.
5 points. The joint distribution of each (Xi, Yi, Zi) can be specified by five parameters: Po = P(X; = 1|2i = 0), P1 = P(Xi = 1|2i = 1), 1Zi Zi , - = = = = = 2 = = = = = = 2 = - qo = P(Yį = 1|2i = 0), 91 = P(Y; = 11Z1 = 1), Zi Zi p=P(Zi = 1). Use these parameters to write down the joint PMF fx,y,z(x, y, z; PO, P1, 90, 91, r) according to which each random vector (Xi, Yi, Zi) is distributed. Specifically, you should write a compact (i.e., not piecewise) expression using the same trick we used in writing down the Bernoulli PMF >
*, = = * 2* = 10 points. Consider a sample (X*, Y*, Z*) from the same distribution, and suppose that we observe that X. = 0 and Y. = 1. Conditional on this observation, what is the probability that * Zn = 1? (Your answer should be written in terms of the exact parameters.) 10 points. Forgetting about the additional sample (X, Y, Z+), compute the MLEs Po, P1, ĝo, ĝi, ř for the corresponding parameters. 5 points. Compute the standard error of each of po, ſ1, ĝo, ĝi, f. * *, 2 2 2 > 2
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5 points. The joint distribution of each (Xi, Yi, Zi) can be specified by five parameters: Po = P(X; = 1|2i = 0), P1 = P(Xi = 1|2i = 1), 1Zi Zi , - = = = = = 2 = = = = = = 2 = - qo = P(Yį = 1|2i = 0), 91 = P(Y; = 11Z1 = 1), Zi Zi p=P(Zi = 1). Use these parameters to write down the joint PMF fx,y,z(x, y, z; PO, P1, 90, 91, r) according to which each random vector (Xi, Yi, Zi) is distributed. Specifically, you should write a compact (i.e., not piecewise) expression using the same trick we used in writing down the Bernoulli PMF >
*, = = * 2* = 10 points. Consider a sample (X*, Y*, Z*) from the same distribution, and suppose that we observe that X. = 0 and Y. = 1. Conditional on this observation, what is the probability that * Zn = 1? (Your answer should be written in terms of the exact parameters.) 10 points. Forgetting about the additional sample (X, Y, Z+), compute the MLEs Po, P1, ĝo, ĝi, ř for the corresponding parameters. 5 points. Compute the standard error of each of po, ſ1, ĝo, ĝi, f. * *, 2 2 2 > 2