Homework Answers
Since it is a single measurement, the mean and variance of the distribution is the same as the value
Thus,
Mean = 
Variance =
Standard deviation = 
Best estimate of the deviation from the true mean = 10
Answer :
C. 10
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17. Assume the measurement we conduct follows Poisson Distribution model. We have a single measurement x...
The Poisson distribution is a useful discrete distribution which can be used to model the number ...
PROBABILITY QUESTION
The Poisson distribution is a useful discrete distribution which can be used to model the number of occur rences of something per unit time. If X is Poisson distributed, i.e. X Poisson(λ), its probability mass function takes the following form: oisson distributed, i.e. X - Assume now we have n identically and independently drawn data points from Poisson(A) :D- {r1,...,Xn Question 3.1 [5 pts] Derive an expression for maximum likelihood estimate (MLE) of λ. Question 3.2 5pts Assume...
If X follows a Poisson distribution with parameter lemda, such that p(x=2)= 9 ( p (x=4)...
If X follows a Poisson distribution with parameter lemda, such that p(x=2)= 9 ( p (x=4) +10 p(x=6) ). Find ( mean+ 3 standard deviation) and (mean - 3 standard deviation). comment on the result.
Suppose that N ~ Poisson(1), and that X has a conditional distribution that depends on N....
Suppose that N ~ Poisson(1), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: • Model M1: X|N = n ~ N(2* n,o2 = 1) • Model M2: X|N = n ~ N(2*n+n2,02 = 1) The only difference between My and M2 is the conditional mean function - the mean function is linear in M1 and quadratic in M2. a If M1 is true: Find E1[X] and V1[X] as...
Question 3 (5101) Suppose that N ~ Poisson(A), and that X has a conditional distribution that...
Question 3 (5101) Suppose that N ~ Poisson(A), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: . Model M: X|N=n~ N(2*n, o2 = 1) • Model M2: X|N=n~ N(2+n + n2,02 = 1) The only difference between M and M2 is the conditional mean function - the mean function is linear in M, and quadratic in M2. a If Mi is true: Find Ej[X] and V1 [X] as...
Question 3 (5101) Suppose that N ~ Poisson(2), and that X has a conditional distribution that...
Question 3 (5101) Suppose that N ~ Poisson(2), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: • Model M: X|N=n~ N(2*n, 02 = 1) • Model M2: X|N=n~ N(2+n + n2,02 = 1) The only difference between M, and M2 is the conditional mean function - the mean function is linear in M, and quadratic in M2. a If M is true: Find E(X) and V. [X] as...
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,...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each oth...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each oth...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
Assume a random variable XX follows a Poisson distribution with a mean μ=3.7μ=3.7. Find P(X≤4) P(X≤4)=
Assume a random variable XX follows a Poisson distribution with a mean μ=3.7μ=3.7. Find P(X≤4) P(X≤4)=
11.2 Let X have the Poisson distribution with parameter 2. a) Determine the MGF of X....
11.2 Let X have the Poisson distribution with parameter 2. a) Determine the MGF of X. Hint: Use the exponential series, Equation (5.26) on page 222 b) Use the result of part (a) to obtain the mean and variance of X. ons, binomial probabilities can -a7k/k!. These quantities are useful The Poisson Distribution From Proposition 5.7, we know that, under certain conditions, binomial be well approximated by quantities of the form e-^1/k!. These in many other contexts. begin, we show...
PROBABILITY QUESTION
The Poisson distribution is a useful discrete distribution which can be used to model the number of occur rences of something per unit time. If X is Poisson distributed, i.e. X Poisson(λ), its probability mass function takes the following form: oisson distributed, i.e. X - Assume now we have n identically and independently drawn data points from Poisson(A) :D- {r1,...,Xn Question 3.1 [5 pts] Derive an expression for maximum likelihood estimate (MLE) of λ. Question 3.2 5pts Assume...
Suppose that N ~ Poisson(1), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: • Model M1: X|N = n ~ N(2* n,o2 = 1) • Model M2: X|N = n ~ N(2*n+n2,02 = 1) The only difference between My and M2 is the conditional mean function - the mean function is linear in M1 and quadratic in M2. a If M1 is true: Find E1[X] and V1[X] as...
Question 3 (5101) Suppose that N ~ Poisson(A), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: . Model M: X|N=n~ N(2*n, o2 = 1) • Model M2: X|N=n~ N(2+n + n2,02 = 1) The only difference between M and M2 is the conditional mean function - the mean function is linear in M, and quadratic in M2. a If Mi is true: Find Ej[X] and V1 [X] as...
Question 3 (5101) Suppose that N ~ Poisson(2), and that X has a conditional distribution that depends on N. There are two possible models for this conditional distribution: • Model M: X|N=n~ N(2*n, 02 = 1) • Model M2: X|N=n~ N(2+n + n2,02 = 1) The only difference between M, and M2 is the conditional mean function - the mean function is linear in M, and quadratic in M2. a If M is true: Find E(X) and V. [X] as...
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,...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
11.2 Let X have the Poisson distribution with parameter 2. a) Determine the MGF of X. Hint: Use the exponential series, Equation (5.26) on page 222 b) Use the result of part (a) to obtain the mean and variance of X. ons, binomial probabilities can -a7k/k!. These quantities are useful The Poisson Distribution From Proposition 5.7, we know that, under certain conditions, binomial be well approximated by quantities of the form e-^1/k!. These in many other contexts. begin, we show...
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