Suppose
X1, X2, X3
are i.i.d. Exp( λ ), and that we observe the realizations
X1 =
1.0, X2...
Suppose
X1, X2, X3
are i.i.d. Exp( λ ), and that we observe the realizations
X1 =
1.0, X2 = 2.0, and
X3 = 3.0. What is the maximum likelihood
estimate of Pr(X1> 2)?
Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1
+ X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y =
(Y1,Y2,Y3)′ using : Multivariate normal distribution
properties.
Let X1,X2,...,Xn be i.i.d. Gamma random variables with
parameters α and λ. The likelihood function is difficult to
differentiate because of the gamma function. Rather than finding the
maximum likelihood estimators, what are the method of moments
estimators of both parameters α and λ?
Suppose X1,···, Xn ∼ Exp(λ) are
independent. What is the distribution of X1/S where S =
X1+X2+···+Xn?
Please show me how to do this without using the property
of chi-squared dist.
Suppose X1,···, Xn ∼ Exp(λ) are
independent. What is the distribution of X1/S where S =
X1+X2+···+Xn?
Please show me how to do this without using the property
of chi-squared dist.
(a) Consider three positive integers, x1, x2, x3, which satisfy
the inequality below: x1 +x2 +x3 =17. (1) Let’s assume each element
in the sample space (consisting of solution vectors (x1, x2, x3)
satisfying the above conditions) is equally likely to occur. For
example, we have equal chances to have (x1, x2, x3) = (1, 1, 15) or
(x1, x2, x3) = (1, 2, 14). What is the probability the events x1
+x2 ≤8occurs,i.e.,P(x1 +x2 ≤8|x1 +x2 +x3 =17andx1,x2,x3 ∈Z+)(Z+...
Suppose we have a random sample of n observations
{x1, x2, x3,…xn}.
Consider the following estimator of µx, the population
mean.
Z = 12x1 +
14x2 +
18x3 +…+
12n-1xn−1 +
12nxn
Verify that for a finite sample size, Z is a biased
estimator.
Recall that Bias(Z) = E(Z) − µx. Write down a
formula for Bias(Z) as a function of n and µx.
Is Z asymptotically unbiased? Explain.
Use the fact that for 0 < r < 1,
limn→∞i=1nri...
Does the input requirement set
V (y) = {(x1, x2, x3) | x1 + min {x2, x3} ≥ 3y, xi ≥ 0
∀ i = 1, 2, 3}
corresponds to a regular (closed and non-empty) input
requirement set?
Does the technology satisfies free disposal? Is the technology
convex?