Question

Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant rate and independently of the time since the last event. For instance, a local government keeping track of the amount of COVID-19 death cases reported each day may notice that an average number of new daily cases is 10. Without loss of generality, we may assume that the number of new daily cases obeys a Poisson distribution with p.m.f. px(x) = e^-??^x / x!. Question 1.1 (2 points) Suppose X1, X2, ..., Xn is a sample from a Poisson distribution with parameter ?. Let X? = 1/n ?i=1^n Xi. Show the derivation of maximum likelihood estimator (MLE) of ?. Question 1.1 (6 points) If X ~ Poi(?), then E[X] = Var(X) = ? (NO NEED TO DERIVE THESE). Suppose X1, X2, ..., Xn is a sample from a Poisson distribution with parameter ?. Write down the Central Limit Theorem and show the derivation of 95% confidence interval of ? in terms of n and X?. Question 1.2 (2 points) Suppose we recorded the number of new daily deaths in the past seven days, and obtained the numbers: 13, 9, 13, 15, 11, 13, and 7. If we assume this sample is independent and identically distributed from a Poisson distribution, then what is the 95% confidence interval of ??

          Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant rate and independently of the time since the last event. For instance, a local government keeping track of the amount of COVID-19 death cases reported each day may notice that an average number of new daily cases is 10. Without loss of generality, we may assume that the number of new daily cases obeys a Poisson distribution with p.m.f. px(x) = e^-??^x / x!.

Question 1.1 (2 points)
Suppose X1, X2, ..., Xn is a sample from a Poisson distribution with parameter ?. Let X? = 1/n ?i=1^n Xi. Show the derivation of maximum likelihood estimator (MLE) of ?.

Question 1.1 (6 points)
If X ~ Poi(?), then E[X] = Var(X) = ? (NO NEED TO DERIVE THESE). Suppose X1, X2, ..., Xn is a sample from a Poisson distribution with parameter ?. Write down the Central Limit Theorem and show the derivation of 95% confidence interval of ? in terms of n and X?.

Question 1.2 (2 points)
Suppose we recorded the number of new daily deaths in the past seven days, and obtained the numbers: 13, 9, 13, 15, 11, 13, and 7. If we assume this sample is independent and identically distributed from a Poisson distribution, then what is the 95% confidence interval of ??

Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant rate and independently of the time since the last event. For instance, a local government keeping track of the amount of COVID-19 death cases reported each day may notice that an average number of new daily cases is 10. Without loss of generality, we may assume that the number of new daily cases obeys a Poisson distribution with p.m.f. px(x) = e^-??^x / x!.

Question 1.1 (2 points)
Suppose X1, X2, ..., Xn is a sample from a Poisson distribution with parameter ?. Let X? = 1/n ?i=1^n Xi. Show the derivation of maximum likelihood estimator (MLE) of ?.

Question 1.1 (6 points)
If X Poi(?), then E[X] = Var(X) = ? (NO NEED TO DERIVE THESE). Suppose X1, X2, ..., Xn is a sample from a Poisson distribution with parameter ?. Write down the Central Limit Theorem and show the derivation of 95% confidence interval of ? in terms of n and X?.

Question 1.2 (2 points)
Suppose we recorded the number of new daily deaths in the past seven days, and obtained the numbers: 13, 9, 13, 15, 11, 13, and 7. If we assume this sample is independent and identically distributed from a Poisson distribution, then what is the 95% confidence interval of ??

Added by Kevin G.

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