Homework 7.pdf

Homework 71.Suppose that X and Y have a continuous joint distribution for which the joint p.d.f.isas follows:f(x,y)=x+y,for 0 x 1,and 0 y 10,otherwise.(1)Find E(Y|X)and V ar(Y|X).(1)Find V arE(Y|X)and EV ar(Y|X).2.Suppose that on a certain examination in advanced mathematics,students from univer-sity A achieve scores that are normally distributed with a mean of 625 and a varianceof 100,and students from university B achieves scores which are normally distributedwith a mean of 600 and a variance of 150.If two students from university A andthree students from university B take this examination,what is the probability thatthe average of the scores of the two students from university A will be greater thanthe average of the scores of the three students form university B?Hint:Determine thedistribution of the difference between the two averages.13.Suppose that a random variable X has a normal distribution,and for every x,theconditional distribution of another random variable Y given X=x is a normal distri-bution with mean ax+b and variance 2,where a,b and 2are constants.Prove thatthe joint distribution of X and Y is a bivariate normal distribution.4.Let X1,X2,.,Xnrepresent a random sample from each of the distributions havingthe following probability density functions:(a)f(x;)=xe/x!,x=0,1,2,.,0 ,zero elsewhere,where f(0;0)=1.(b)f(x;)=x1,0 x 1,0 ,zero elsewhere.(c)f(x;)=12e|x|,x ,0 and 2 0If X1,X2,.,Xnis a random sample from this distribution,find the maximum likeli-hood estimators of 1and 2.2