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Show that the following distributions belong to the exponential family. Find the natural parameter θ, scale...
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,...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t,...
Show that the family of distributions GAM , к): > 0, к > 0 belongs to the regular exponential cla...
Show that the family of distributions GAM , к): > 0, к > 0 belongs to the regular exponential class, and use this information to find com plete sufficient statistics based on a random sample Xi,... , X, Let Xi,... , Xn be a random sample from a Bernoulli distribution, Xi BIN(1,p):0<p1. EX)-Σ x, n(n -1) Verify that T- is an unbiased estimator of p Then show that T is a UMVUE of p2
Show that the family of distributions...
Find the 60th percentile of the following distributions: (a) Exponential with mean θ (b) Continuous uniform...
Find the 60th percentile of the following distributions: (a) Exponential with mean θ (b) Continuous uniform on [1,5] (c) f (x)= (x+1)/2 ,−1< x <1
1. Consider a GLM (generalised linear model) for a Poisson random sample Y1,. .. , Y, with \Vi each Yi having a pdf or...
1. Consider a GLM (generalised linear model) for a Poisson random sample Y1,. .. , Y, with \Vi each Yi having a pdf or pmf f(y; A;) = i= 1, . .. ,n. Yi = 0, 1,2, -..; ^; > 0; Y;! Note that the pdf from an exponential family has the following general form b(0) + c(y, a(o) y0 exp f(y; 0, 6) = Suppose the linear predictor of the GLM is n = a+Bxi, with (a,B) being the...
4. For known k 2, show that the negative binomial distribution with probability mass function, f(yk, )-y+k-1 ;y0,1,2, b...
4. For known k 2, show that the negative binomial distribution with probability mass function, f(yk, )-y+k-1 ;y0,1,2, belongs to the exponential family of distributions. Find the natural parameter for this dis tribution
Problem 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity...
Problem 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity λ-1. Given L-u, a random variable Y has the Poisson distribution with parameter - u. 1. Derive the marginal distribution of Y and evaluate probabilities, PY=n] , for n = 0,1,2, 2. Find the expectation of Y, that is E Y 3. Find the variance of Y, that is Var Y
4 10 pts. Let X1 X2 be a random sample from the exponential distribution with parameter θ What is the mgf of Y = X1 + X2? a) (4 pts.+) Find E(Y-E(X1 + X2] using the mgf. For 2 more points on test 2:...
4 10 pts. Let X1 X2 be a random sample from the exponential distribution with parameter θ What is the mgf of Y = X1 + X2? a) (4 pts.+) Find E(Y-E(X1 + X2] using the mgf. For 2 more points on test 2: How is Y distributed?
4 10 pts. Let X1 X2 be a random sample from the exponential distribution with parameter θ What is the mgf of Y = X1 + X2? a) (4 pts.+) Find E(Y-E(X1...
1. Let {x, t,f 0) and {Yǐ.12 0) be independent Poisson processes,with rates λ and 2A, respectivel...
1. Let {x, t,f 0) and {Yǐ.12 0) be independent Poisson processes,with rates λ and 2A, respectively. Obtain the conditionafdistributiono) Moreover, find EX Y X2t t given Yt-n, n = 1,2. 2, (a) Let T be an exponential random variable with parameter θ. For 12 0, compute (b) When Amelia walks from home to work, she has to cross the street at a certain point. Amelia needs a gap of a (units of time) in the traffic to cross the...
: Let Yi, ½' . . . , Yn be an iid random sample from an exponential distribution with parameter where θ > 0. Here each Y, represents the lifetime of the ith battery, while θ represents the the...
: Let Yi, ½' . . . , Yn be an iid random sample from an exponential distribution with parameter where θ > 0. Here each Y, represents the lifetime of the ith battery, while θ represents the theoretical average lifetime. The pdf of each Y, is therefore given by fy (y) ei-1,2,...,n Consider the empirical average lifetime of the sample of n batteries given by Let a E R be a nonnegative real number. Consider the event A, defined...
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,...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t,...
Show that the family of distributions GAM , к): > 0, к > 0 belongs to the regular exponential class, and use this information to find com plete sufficient statistics based on a random sample Xi,... , X, Let Xi,... , Xn be a random sample from a Bernoulli distribution, Xi BIN(1,p):0<p1. EX)-Σ x, n(n -1) Verify that T- is an unbiased estimator of p Then show that T is a UMVUE of p2
Show that the family of distributions...
1. Consider a GLM (generalised linear model) for a Poisson random sample Y1,. .. , Y, with \Vi each Yi having a pdf or pmf f(y; A;) = i= 1, . .. ,n. Yi = 0, 1,2, -..; ^; > 0; Y;! Note that the pdf from an exponential family has the following general form b(0) + c(y, a(o) y0 exp f(y; 0, 6) = Suppose the linear predictor of the GLM is n = a+Bxi, with (a,B) being the...
4. For known k 2, show that the negative binomial distribution with probability mass function, f(yk, )-y+k-1 ;y0,1,2, belongs to the exponential family of distributions. Find the natural parameter for this dis tribution
Problem 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity λ-1. Given L-u, a random variable Y has the Poisson distribution with parameter - u. 1. Derive the marginal distribution of Y and evaluate probabilities, PY=n] , for n = 0,1,2, 2. Find the expectation of Y, that is E Y 3. Find the variance of Y, that is Var Y
4 10 pts. Let X1 X2 be a random sample from the exponential distribution with parameter θ What is the mgf of Y = X1 + X2? a) (4 pts.+) Find E(Y-E(X1 + X2] using the mgf. For 2 more points on test 2: How is Y distributed?
4 10 pts. Let X1 X2 be a random sample from the exponential distribution with parameter θ What is the mgf of Y = X1 + X2? a) (4 pts.+) Find E(Y-E(X1...
1. Let {x, t,f 0) and {Yǐ.12 0) be independent Poisson processes,with rates λ and 2A, respectively. Obtain the conditionafdistributiono) Moreover, find EX Y X2t t given Yt-n, n = 1,2. 2, (a) Let T be an exponential random variable with parameter θ. For 12 0, compute (b) When Amelia walks from home to work, she has to cross the street at a certain point. Amelia needs a gap of a (units of time) in the traffic to cross the...
: Let Yi, ½' . . . , Yn be an iid random sample from an exponential distribution with parameter where θ > 0. Here each Y, represents the lifetime of the ith battery, while θ represents the theoretical average lifetime. The pdf of each Y, is therefore given by fy (y) ei-1,2,...,n Consider the empirical average lifetime of the sample of n batteries given by Let a E R be a nonnegative real number. Consider the event A, defined...
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