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4. For known k 2, show that the negative binomial distribution with probability mass function, f(yk, )-y+k-1 ;y0,1,2, b...
Show that the following distributions belong to the exponential family. Find the natural parameter θ, scale...
Show that the following distributions belong to the exponential family. Find the natural parameter θ, scale parameter p and convex function b(9). Also find the E(Y) and Var(Y) as functions of the natural parameter. Specify the canonical link functions 1. Exponential distribution Bxp ), f(y:λ) λe-Ag. Binomial distribution known; f(y: π- C)π"(1-π)n-y, where n is 2. Bin(n,π). 3. Poisson distribution Pois(A), f(y:A)-e
He second form for one-parameter exponential family distributions, introduced during lecture 09.1...
he second form for one-parameter exponential family distributions, introduced during lecture 09.1, was Jy (y | θ) = b(y)ec(0)t(y)-d(0) Let η = c(0). If c is an invertible function, we can rewrite (1) as where η is called the natural, or canonical, parameter and K(n) = d(C-1(n)). Expression (2) is referred to as the canonical representation of the exponential family distribution (a) Function κ(η) is called the log-normalizer: it ensures that the distribution fy(y n) integrates to one. Show that,...
The following joint probability distribution is given. 1. Find k such that the given function demonstrates...
The following joint probability distribution is given. 1. Find k
such that the given function demonstrates the PDF. 2. Find Marginal
distributions. 3. Evaluate ?(? < ? < 0) 4. Find the
correlation coefficient between X and Y having the joint density
functions:(.) ?(?,?) = {???2+?2 ??? ?2 + ?2 < 4 0
?????h???
Question 2. (20 pts.) The following joint probability distribution is given. 1. Find k such that the given function demonstrates the PDF. 2. Find Marginal distributions....
(3.4) This question is about a continuous probability dis- tribution known as the exponential distribution Let...
(3.4) This question is about a continuous probability dis- tribution known as the exponential distribution Let x be a continuous random variable that can take any value x 20. A quantity is said to be exponen- tially distributed if it takes values between r and r + dr with probability where A and A are constants. (a) Find the value of A that makes P() a well- defined continuous probability distribution so that Jo o P(x) dx = 1 (b)...
1. X,,x2,..., X, is a random sample from a Poisson (0) distribution with probability mass function...
1. X,,x2,..., X, is a random sample from a Poisson (0) distribution with probability mass function 0*e f(x) = x=0,1,..., 0 >0. x! (1) Write Poisson (0) as an exponential family of the form fo(x) = exp{c(0)T(x)-v (0)}h(x) State what c(0), 7(x), and y (@) are. (ii) a. Prove that for the exponential family given in (i), E[T(X)]=y'(c). b. Hence find the mean of the Poisson (0) distribution. [3] [6] [2] 21 (iii) Show that for the Poisson (0) distribution,...
5. Let f(t) be the probability density function, and F(t) be the corresponding cumulative f(t) distribution...
5. Let f(t) be the probability density function, and F(t) be the corresponding cumulative f(t) distribution function. Define the hazard function h(t) Show that if X is an 1-F(t): exponential random variable with parameter 1 > 0, then its hazard function will be a constant h(t) = 1 for all t > 0. Think of how this relates to the memorylessness property of exponential random variables.
Let the random variable Y have the following probability distribution y 2 4 6 P(Y=y) 4/k...
Let the random variable Y have the following probability distribution y 2 4 6 P(Y=y) 4/k 1/k 5/k find the value of k. find the moment-generating function of Y find Var(Y) using the moment generating function let W= 2Y-Y^2 +e^2*Y+7. find E(W)
The random variable X has probability density function k(x25x-4) 1<x<4 otherwise -{ f(x) 1. Show thatk....
The random variable X has probability density function k(x25x-4) 1<x<4 otherwise -{ f(x) 1. Show thatk. (5pts) Find 2. Е (X), (5pts) 3. the mode of X, (5pts) 4. the cumulative distribution function F(X) for all x. (5pts) 5. Evaluate P(X < 2.5). (5pts) 6. Deduce the value of the median and comment on the shape of the distribution (10pts)
(a) Consider a Poisson distribution with probability mass function: еxp(- в)в+ P(X = k) =- k!...
(a) Consider a Poisson distribution with probability mass function: еxp(- в)в+ P(X = k) =- k! which is defined for non-negative values of k. (Note that a numerical value of B is not provided). Find P(X <0). (i) (4 marks) Find P(X > 0). (ii) (4 marks) Find P(5 < X s7). (ii) (4 marks) 2.
Let F (x, y, 2) =zi+xj+yk. Find the divergence of F. a. -2 b. -1 c....
Let F (x, y, 2) =zi+xj+yk. Find the divergence of F. a. -2 b. -1 c. 0 d. 1 e. 2 f. 3 g. 4 h. 5
Show that the following distributions belong to the exponential family. Find the natural parameter θ, scale parameter p and convex function b(9). Also find the E(Y) and Var(Y) as functions of the natural parameter. Specify the canonical link functions 1. Exponential distribution Bxp ), f(y:λ) λe-Ag. Binomial distribution known; f(y: π- C)π"(1-π)n-y, where n is 2. Bin(n,π). 3. Poisson distribution Pois(A), f(y:A)-e
he second form for one-parameter exponential family distributions, introduced during lecture 09.1, was Jy (y | θ) = b(y)ec(0)t(y)-d(0) Let η = c(0). If c is an invertible function, we can rewrite (1) as where η is called the natural, or canonical, parameter and K(n) = d(C-1(n)). Expression (2) is referred to as the canonical representation of the exponential family distribution (a) Function κ(η) is called the log-normalizer: it ensures that the distribution fy(y n) integrates to one. Show that,...
The following joint probability distribution is given. 1. Find k
such that the given function demonstrates the PDF. 2. Find Marginal
distributions. 3. Evaluate ?(? < ? < 0) 4. Find the
correlation coefficient between X and Y having the joint density
functions:(.) ?(?,?) = {???2+?2 ??? ?2 + ?2 < 4 0
?????h???
Question 2. (20 pts.) The following joint probability distribution is given. 1. Find k such that the given function demonstrates the PDF. 2. Find Marginal distributions....
(3.4) This question is about a continuous probability dis- tribution known as the exponential distribution Let x be a continuous random variable that can take any value x 20. A quantity is said to be exponen- tially distributed if it takes values between r and r + dr with probability where A and A are constants. (a) Find the value of A that makes P() a well- defined continuous probability distribution so that Jo o P(x) dx = 1 (b)...
1. X,,x2,..., X, is a random sample from a Poisson (0) distribution with probability mass function 0*e f(x) = x=0,1,..., 0 >0. x! (1) Write Poisson (0) as an exponential family of the form fo(x) = exp{c(0)T(x)-v (0)}h(x) State what c(0), 7(x), and y (@) are. (ii) a. Prove that for the exponential family given in (i), E[T(X)]=y'(c). b. Hence find the mean of the Poisson (0) distribution. [3] [6] [2] 21 (iii) Show that for the Poisson (0) distribution,...
5. Let f(t) be the probability density function, and F(t) be the corresponding cumulative f(t) distribution function. Define the hazard function h(t) Show that if X is an 1-F(t): exponential random variable with parameter 1 > 0, then its hazard function will be a constant h(t) = 1 for all t > 0. Think of how this relates to the memorylessness property of exponential random variables.
The random variable X has probability density function k(x25x-4) 1<x<4 otherwise -{ f(x) 1. Show thatk. (5pts) Find 2. Е (X), (5pts) 3. the mode of X, (5pts) 4. the cumulative distribution function F(X) for all x. (5pts) 5. Evaluate P(X < 2.5). (5pts) 6. Deduce the value of the median and comment on the shape of the distribution (10pts)
(a) Consider a Poisson distribution with probability mass function: еxp(- в)в+ P(X = k) =- k! which is defined for non-negative values of k. (Note that a numerical value of B is not provided). Find P(X <0). (i) (4 marks) Find P(X > 0). (ii) (4 marks) Find P(5 < X s7). (ii) (4 marks) 2.
Let F (x, y, 2) =zi+xj+yk. Find the divergence of F. a. -2 b. -1 c. 0 d. 1 e. 2 f. 3 g. 4 h. 5
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