Let Z1, Z2, . . . , Zn be independent and identically
distributed as standard normal random variables. Prove the
distribution of ni=1 Zi2 ∼ χ2n.
Thanks!
1. Suppose that Z1,Z2 are independent standard normal random
variables. Let Y1 = Z1 − 2Z2, Y2 = Z1 − Z2.
(a) Find the joint pdf fY1,Y2(y1,y2). Don’t use the change of
variables theorem – all of that work has already been done for you.
Instead, evaluate the matrices Σ and Σ−1, then multiply the
necessary matrices and vectors to obtain a formula for
fY1,Y2(y1,y2) containing no matrices and no vectors.
(b) Find the marginal pdf fY2 (y2). Don’t use...
let r1 and r2 be the relations represented as r1 (ABC)
and r2 (ADE) .Assume the corresponding domains of both the tables
are same.r1 has 2000 tuples and r2 has 4500 tuples
1.common tuples between r1 and r2 are 500, what would be the
resultant number of tuples for r1-r2, justify your answer
2.assuming 500 as the common tuples between r1 and r2,what is the
maximum number of tuples that results in ]] A( r1) U ]] A r2.
justify...
Let the random variable Z follow a standard normal distribution,
and let Z1 be a possible value of Z that is representing the 90th
percentile of the standard normal distribution. Find the value of
Z1.
Suppose Z1 and Z2 are two standard norm random variables. In
addition suppose cov(Z1,Z2)=p.
Show (Z1-pZ2)/sqrt(1-p^2)and Z2 are standard normally
distributed
Show (Z1-pZ2) )/sqrt(1-p^2)and Z2 are independent. Hint : Two
random normal variables are independent as long as they are
uncorrelated
Show (Z1^2+Z2^2-2pZ1Z2)/(1-p^2) is Chai square
distribution
With reference to equations (4.2)
and (4.3), let Z1 = U1 and Z2 =
−U2 be independent, standard normal variables.
Consider the polar coordinates of the point (Z1,
Z2), that is,
A2 = Z2 + Z2
and φ =
tan−1(Z2/Z1).
1 2
(a) Find the joint density of A2 and
φ, and from the result, conclude that A2 and
φ are independent random variables, where A2 is a
chi-
squared random variable with 2 df, and φ is uniformly
distributed...
2. Let X1, . . . , Xn be a random sample from the distribution
with pdf given by fX(x;β) = β 1(x ≥ 1).
xβ+1
(a) Show that T = ni=1 log Xi is a sufficient statistic for β.
Hint: Use
n1n1n=exp log=exp −logxi .i=1 xi i=1 xi i=1
(b) Find the pdf of Y = logX, where X ∼ fX(x;β).
(c) Find the distribution of T . Hint: Identify the distribution of
Y and use mgfs.
(d) Find...
Let X1, . . . , Xn be a random sample from a uniform
distribution on the interval [a, b]
(i) Find the moments estimators of a and b.
(ii) Find the MLEs of a and b.