(7) Find the distribution of the so-called "extreme value" density function f(x) = exp(-r-e-r) for x R. (7) Find the distribution of the so-called "extreme value" density functio...
Problem(2) (5 points) A continuous distribution has density function k sin(r); 0rST 0; f(x) = otherwise. (a) Find the numerical value of k so that f(r) is a density function. (b) Find E[X] (c) Find E[X2. (d) Find Var X] Problem(2) (5 points) A continuous distribution has density function k sin(r); 0rST 0; f(x) = otherwise. (a) Find the numerical value of k so that f(r) is a density function. (b) Find E[X] (c) Find E[X2. (d) Find Var X]
In question 5, f(x) = λ*exp(-λx), for x greater or equal to 0, and zero otherwise. 9. Let X have an exponential distribution with λ = 1 (see Question 5), and let Y = log(X). Find the probability density function of Y. Where is the density non-zero? Note that in this course, log refers to the log base e, or natural log, often symbolized In. The distribution of Y is called the (standard) Gumbel, or extreme value distribution. 2
(7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi , . . . , X,.), V=min(X1, ,X,). (a) Find the distribution function and the density function of U and of V (b) Show that the joint density function of U and V is fe,y(u, u)= n(n-1)/(u)/(v)[F(v)-F(u)]n-1, ifu < u. (7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi...
Find the median of exponential distribution with probability density function f(x) = * e * P -2
Let X1,...X be i.i.d with density f()(1/0)exp(-/0) for r >0 and 0> 0. a. Find the pitman estimator of 0 b. Show that the pitman estimator has smaller risk than the UMVUE of when the loss function is (t-0)2 02 Suppose C. f(x)= 0exp(-0x) and that 0 has a gamma prior with parameters a and p, find the Bayes estimator of 0 d. Find the minimum Bayes risk e. Find the minimax estimator of 0 if one exists. 1 Let...
QUESTION6 (a) The three-parameter gamma distribution has the probability density function x (r)- exp (r-c)-1,x> Derive the mean of the distribution. (b) Ifx beta I (m, n) show thatY ax +b(l-X) has the four-parameter beta distribution with parameters a, b, m and n.
Suppose X follows a distribution with density function: f(x)-10.7kz2 031 otherwise (Note: for this question you can enter your answer in decimals as well as fractions) 1. What must the value of k be so that f(x)is a probability density function? Submit Answer Tries 0/3 2. Find the probability P(0.40.8) Submit Answer Tries 0/3 . Find median of the distribution of X Submit Answer Tries 0/3 4. Find E(Xx) Submit Answer Tries 0/3 5. Find Var(x) Submit Answer Tries 0/3...
Let f(x,y) = exp(-x) be a probability density function over the plane. Find the probabilities: Parta)P( X2 + y2 <a), a > 0, Part b)P(x2 + y2 <a), a > 0.
10. (10 points) A function f : R2 + R is called a probability density function on D CR if (6) f(, y) 0 for all (x, y) E D and (i) SD. f(x,y)dA= 1. ſk(1 – 22 – y2) 22 + y2 <1 (a) For what constant k is the function f(z,y) a prob- 12 + y2 > 1 ability density function? Note that D= {(1, Y) ER? : x2 + y² <1}, the closed unit disk in R2...
A continuous random variable Y has density function f(y) = f'(y) = 2 · exp[-4. [y] defined for -00 < y < 0. Evaluate the cumulative distribution function for Y Consider W = |Y| and find its C.D.F. and density Determine expected value, E [Y] Derive variance, Var [Y]