2. A computer network experiences 4 disconnections in one minute, 2 in another minute and 7...
2. A computer network experiences 4 disconnections in one minute, 2 in another minute and 7 in another minute. Assume the number N of disconnections in one minute follows a Poisson distribution with parameter Λ, so Pr(N = k) = e −ΛΛ k k! . (a) Given this data, a likelihood function L for the parameter Λ. (b) Find all the critical points of L. (c) Find the maximum likelihood estimator Λ.
Homework Answers
Likelihood function, 
where 
L=
implies
is a point of maximum
M.L.E of
according to the above data
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2. A computer network experiences 4 disconnections in one
minute, 2 in another minute and 7...
A computer network experiences 4 disconnections in one minute, 2 in another minute and 7 in anoth...
A computer network experiences 4 disconnections in one minute, 2 in another minute and 7 in another minute. Assume the number N of disconnections in one minute follows a Poisson distribution with parameter Λ, so Pr(N=k)= ((e^-/\ )*(/\^k)) divided by k! (a) Given this data, a likelihood function L for the parameter Λ. (b) Find all the critical points of L. (c) Find the maximum likelihood estimator Λ.
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,...
The number of medical emergency calls per hour has a Poisson distribution with parameter λ. Calls...
The number of medical emergency calls per hour has a Poisson distribution with parameter λ. Calls received at different hours are considered to be independent. Emergency calls X1 ,…, Xn for n consecutive hours has the same parameter λ. a) What is the distribution of Sn = ∑ Xi ? b) Provide Normal approximation for the distribution of Sn . c) Provide maximum likelihood estimation of λ. Calculate variance and bias of MLE. d) Calculate Fisher information and efficiency of...
4 Mary is a waitress in a city centre restaurant. She receives tips from customers at an average rate of λ per hour. During a particular eight hour shift, she receives 5 tips. We wish to estimate λ....
4 Mary is a waitress in a city centre restaurant. She receives tips from customers at an average rate of λ per hour. During a particular eight hour shift, she receives 5 tips. We wish to estimate λ. a (2 marks) Let X be the number of tips Mary receives in an 8 hour period. If tips are received according to a Poisson process with rate A tips per hour, state the distribution of X, with parameters b (5 marks)...
Mary is a waitress in a city centre restaurant. She receives tips from customers at an average rate of λ per hour. Dur...
Mary is a waitress in a city centre restaurant. She receives tips from customers at an average rate of λ per hour. During a particular eight a (2 marks) Let X be the number of tips Mary receives in an 8 hour period. If tips are received according to a Poisson process with rate λ tips per hour, state the distribution of X, with parameters b (5 marks) Write down the likelihood function, L(X; 5) for this problem. Remember to...
This question uses a discrete probability distribution known as the Poisson distribution. A discrete random variable...
This question uses a discrete probability distribution known as the Poisson distribution. A discrete random variable X follows a Poisson distribution with parameter λ if Pr(X = k) = Ake-A ke(0, 1,2, ) k! You are a warrior in Peter Jackson's The Hobbit: Battle of the Five Armies. Because Peter decided to make his battle scenes as legendary as possible, he's decided that the number of orcs that will die with one swing of your sword is Poisson distributed (lid)...
Show all working clearly. Thank you. 1. In this question, X is a continuous random variable...
Show all working clearly. Thank you.
1. In this question, X is a continuous random variable with density function (x)a otherwise where ? is an unknown parameter which is strictly positive. You wish to estimate ? using observations X1 , . …x" of an independent random sample XI…·X" from X Write down the likelihood function L(a), simplifying your answer as much as possi- ble 2 marks] i) Show that the derivative of the log likelihood function (a) is 4 marks]...
Suppose we are trying to model the probability q of winning the lottery using a geometric...
Suppose we are trying to model the probability q of winning the lottery using a geometric distribution. Suppose we have one sample, where a person wins the lottery the second time he plays. (a) Find a likelihood function L for the parameter q. (b) Find all critical points of L (c) Find the maximum likelihood estimator ˆq. (d) Compute the upper limit ph of the 95% confidence interval for q. (e) Compute the lower limit pl of the 95% confidence...
Q2 (15%): Suppose that the number of items one purchased online last month Yi depended on...
Q2 (15%): Suppose that the number of items one purchased online last month Yi depended on one’s salary Xi follows a Poisson distribution Yi ∼ Poisson(β0 + β1Xi). We have n pairs of independent observations (x1, y1),(x2, y2), . . . ,(xn, yn) so that Yi are mutually independent. Please use Newton’s method to find the maximum likelihood estimation (MLE) for (β0, β1). Hint: Xis are fixed and Yis are random. Please first write out the probability mass function for...
4. Let X be a random variable that describes the annual counts of tropical cyclones in...
4. Let X be a random variable that describes the annual counts of tropical cyclones in the North Atlantic. Assume that X1,..., X, is a random sample that describes the counts of tropical cyclones in the North Atlantic during n years and assume they are distributed according to a geometric distribution with probability parameter 8 and p.f. given by fxjex | ) = (1 - 0)*-11{1,2,...,x), 0<O<1. (a) Write the statistical model. (b) Find the maximum likelihood estimator of 0....
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,...
4 Mary is a waitress in a city centre restaurant. She receives tips from customers at an average rate of λ per hour. During a particular eight hour shift, she receives 5 tips. We wish to estimate λ. a (2 marks) Let X be the number of tips Mary receives in an 8 hour period. If tips are received according to a Poisson process with rate A tips per hour, state the distribution of X, with parameters b (5 marks)...
Mary is a waitress in a city centre restaurant. She receives tips from customers at an average rate of λ per hour. During a particular eight a (2 marks) Let X be the number of tips Mary receives in an 8 hour period. If tips are received according to a Poisson process with rate λ tips per hour, state the distribution of X, with parameters b (5 marks) Write down the likelihood function, L(X; 5) for this problem. Remember to...
This question uses a discrete probability distribution known as the Poisson distribution. A discrete random variable X follows a Poisson distribution with parameter λ if Pr(X = k) = Ake-A ke(0, 1,2, ) k! You are a warrior in Peter Jackson's The Hobbit: Battle of the Five Armies. Because Peter decided to make his battle scenes as legendary as possible, he's decided that the number of orcs that will die with one swing of your sword is Poisson distributed (lid)...
Show all working clearly. Thank you.
1. In this question, X is a continuous random variable with density function (x)a otherwise where ? is an unknown parameter which is strictly positive. You wish to estimate ? using observations X1 , . …x" of an independent random sample XI…·X" from X Write down the likelihood function L(a), simplifying your answer as much as possi- ble 2 marks] i) Show that the derivative of the log likelihood function (a) is 4 marks]...
4. Let X be a random variable that describes the annual counts of tropical cyclones in the North Atlantic. Assume that X1,..., X, is a random sample that describes the counts of tropical cyclones in the North Atlantic during n years and assume they are distributed according to a geometric distribution with probability parameter 8 and p.f. given by fxjex | ) = (1 - 0)*-11{1,2,...,x), 0<O<1. (a) Write the statistical model. (b) Find the maximum likelihood estimator of 0....
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