Find the Maximum Likelihood estimator of the parameter p?
Q: d) Calculate the maximum likelihood estimate for o2 using this data.
A: only D part
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A: It is widely used in estimation theory . It is very useful in statistics.
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A: ANSWER: For the given data,
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A: The given statement is true.
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- If the random variable X follows the uniform distribution U= (0,1) What is the distribution of the random variable Y= -2lnX. Show its limits.A single observation of a random variable having a hypergeometric distribution with N = 7 and n = 2 is used to test the null hypothesis k = 2 against the alternative hypothesis k = 4. If the null hypothesis is rejected if and only if the value of the random variable is 2, find the power of the test.Suppose a random variable Y uniformly distributed over the interval [2, 7]. 1. What is f(y)?
- Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]A t-distributed random variable T with degrees of freedom df = 5 falls within an interval of values centered around zero, that is the interval -t to t, with probability 0.6. Calculate the value of t that defines that interval.If X1, X2, ... , Xn are independent random variables having identical Bernoulli distributions with the param-eter θ, then X is the proportion of successes in n trials, which we denote by ˆ . Verify that(a) E()ˆ = θ;(b) var()ˆ = θ (1 − θ )n .
- Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?If X1,X2,...,Xn constitute a random sample of size n from a gamma population with α =2, use the method of maximum likelihood to find a formula for estimating β.Suppose the distribution of some random variable X is modeled as Negative binomial with parameters, p and r, where p is the probability of success and r is the total number of success over the trials. Find the method of moment (MM) estimators for (p,r), in closed algebraic form expressions based on a random sample of size n, X1, X2,...,Xn.
- Chebyshev’s theorem provides that for a random variable X with anyprobability distribution and a positive constant k greater than or equalto 1, the inequality P (|X − μ| ≥ kσ) ≤ 1k2 holds. This signifies thatthe probability of X deviating from its mean by at least k standarddeviations is at most 1k2 .a. Determine the upper limit for k being 2, 3, 4, 5, and 10.b. Calculate the mean μ and standard deviation σ for the followingdistribution. Then, assess P (|X − μ| ≥ kσ) for the given k valuesand compare these to the upper bounds found previously.x 0 1 2 3 4 5 6p(x) .1 .15 .2 .25 .06 .042 c. Assume X can take on the values -1, 0, and 1 with probabilities118 , 89 , and 118 respectively. Estimate P (|X − μ| ≥ 3σ), and relateit to the corresponding upper limit.d. Propose a probability distribution where P (|X − μ| ≥ 5σ) is 0.04.A random sample of size n is to be used to test thenull hypothesis that the parameter θ of an exponentialpopulation equals θ0 against the alternative that it doesnot equal θ0.(a) Find an expression for the likelihood ratio statistic.(b) Use the result of part (a) to show that the criticalregion of the likelihood ratio test can be written as x · e−x/θ0 F KIf X1 and X2 are independent random variables hav-ing the geometric distribution with the parameter θ, show that Y = X1 + X2 is a random variable having the neg-ative binomial distribution with the parameters θ and k = 2.