Let U and V be independent Uniform[0, 1] random variables. (a) Calculate E(Uk) where k > 0 is some fixed constant (b) Calculate E(VU)
How can I prove this? 2. (one point) Show that for any three events A, B, and C with P(C) >0, P(A U B|C) = P(A|C) + P(BIC) – P(AN B|C)
3. A random variable X has probability density function f(x) (a-1)2-α for x > 1. (a) For independent observations In show that the log-likelihood is given by, (b) Hence derive an expression for the maximum likelihood estimate for α. (c) Suppose we observe data such that n 6 and Σ61 log(xi) 12. Show that the associated maximum likelihood estimate for α is given by α = 1.5.
Find the Laplace transform of f(0) = 1, for 0 <t<1 5, for 1<t<2. e-l for t > 2
1. Suppose that Y ∼ Gamma(α, β) and c > 0 is a constant. (a) Derive the density function of U = cY. (b) Identify the distribution of U as a standard distribution. Be sure to identify any parameter values. (c) Can you find the distribution of U using MGF method also? I. Suppose that Y ~ Gamma(α, β) and c > 0 is a constant. (a) Derive the density function of U cY. (b) Identify the distribution of U...
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
Let U., Un be independent, identically distributed Uniform random variables with (continu- ous) support on (0, b), where b >0 is a parameter. Define the random variable Y :--Σίι log(U), where log is the natural logarithm function. De- termine the probability density function (pdf) p(y; b of Y by explicitly computing it.
2. (15 marks) Consider the following program: >> Precondition: x and y E Z. Postcondition: Return the sum x + y. add(x, y): 1. if x == 0: 2. return y 3. elif x > 0: 4. return add(x - 1, y) + 1 5. else: 6. return add(x + 1, y) - 1 Prove that this program is correct in terms of its specification.
Suppose that a consumer's utility function is u(x, y) = x, defined for all bundles such that x > 0 and y > 0. For any given positive level of utility, the corresponding indifference curve is O strictly convex. a vertical line. O a horizontal line. a line that is neither horizontal nor vertical.
Let U and V be independent Uniform(0, 1) random variables. (a) Calculate E(Uk) where k> 0 is some fixed constant. (b) Calculate E(VU).