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- Let Y be a binomial random variable with n = 10 and p = 0.3. (a) P(3 < Y < 5) = P(3 ≤ Y < 5) = (b) P(3 < Y ≤ 5) = P(3 ≤ Y ≤ 5) =X denotes a binomial random variable with parameters n and p. For each exercise, indicate which area under the appropriate normal curve would be determined to approximate the specified binomial probability. P(X ≤ 4)A random variable has poisson distribution such that P (X = 3) = P (X= 4). Find p (6).
- Suppose that X1,..., Xn are iid random variables with P(X; = 0) = 1 P(X; = 1) = 2 P(X; = 2) = (1 – 0) 3 1 P(X; = 3) = -(1 – 0), %3D | where 0 < 0 < 1 is the parameter of interest. a) Find the maximum likelihood estimator. Show that its asymptotic variance is 0(1 – 0) n b) Compare this variance to the variance of the method of moments estimator. (c) Give the asymptotic distribution of the maximum likelihood estimator. (d) Using the maximum likelihood estimator, what is the form of an approximate 95% confidence interval for 0?For a random variable, its hazard function also referred to as the instantaneous failure rate is defined as the instantaneous risk (conditional probabilty) that an event of interest will happen in a narrow span of time duration. For a discrete random variable X, its hazard function is defined by the formula hX(k) =P(X=k+ 1|X > k) =pX(k)1−FX(k). For a Poisson distribution with λ= 4.2, find hX(k) and use R to plot the hazard function.If V1, V2, ... , Vn1 and W1, W2, ... , Wn2 are inde-pendent random samples of sizes n1 and n2 from normal populations with the means μ1 and μ2 and the common variance σ2, find maximum likelihood estima-tors for μ1,μ2, and σ2.
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