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- Suppose that X follows a poisson distribution with parameter λ=1.416 . P(X≥0)≅?If X1 and X2 constitute a random sample of size n = 2from a Poisson population, show that the mean of thesample is a sufficient estimator of the parameter λ.Suppose X, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?
- If X1, X2, ... , Xn constitute a random sample from anormal population with μ = 0, show that ni=1X2inis an unbiased estimator of σ2.If the number X of particles emitted during a 1-hour period from a radioactive source has a poisson distribution with parameter equal to 4 and that the probability that any emitted is recorded is p=0.9 find the probability distribution of the number Y of the particles recorded in a 1-hour and hence the probability that no particle is recordedSuppose that it is known that in a certain community 30% of the households have a micro-wave ovens. Fora a simple random sample of 19 households selected from this population find the standard deviation.
- A company has 9000 arrivals of Internet traffic over a period of 18,050 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)= (μ^x • e^−μ) / x! to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?Suppose that X has a Poisson distribution with h = 80. Find theprobability P(x is less than or equal to 80)Suppose the distribution of the time $X$ (in hours) spent by students at a certain university on a particular project is gamma with parameters $\alpha=50$ and $\beta=2 .$ Because $\alpha$ is large, it can be shown that $X$ has approximately a normal distribution. Use this fact to compute the approximate probability that a randomly selected student spends at most 125 hours on the project.
- A company has 8000 arrivals of Internet traffic over a period of 17,460 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)= μx•e−μ x! to find the probability of exactly 3 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?Show that the mean of a random sample of size n from an exponential population is a minimum variance unbi-ased estimator of the parameter θ.Let X1, X2, …, Xn be a random sample from the Normal distribution N() (a) Using method of moments to estimate the parameters and . (b) Are those estimators unbiased?