X1 +x2 + . +Xn Random variables X1,- Xn are independent identically distributed: Let $ be the arithmetic mean of the observed values: Assume that each Xi is a discrete random variable following a geometric PMF with pa- rameter p given by Pxi (xi) = (1 ~p)x = p. Find the maximum likelihood estimator of p in terms of s. [10 pts] 2. Assume that each Xi instead is a continuous random variable following an exponential PDF with parameter 0 given by Oxi 3 @e- Xi > 0 fx; (xi) otherwise: Find the maximum likelihood estimator of 0 in terms of s. [10 pts]
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For the geometric PMF case, we have the likelihood function: L(p) = P(X1, X2, ..., Xn | p) = P(X1 | p) * P(X2 | p) * ... * P(Xn | p) Since the random variables are independent and identically distributed, we can write: L(p) = (1-p)^(x1-1) * p * (1-p)^(x2-1) * p Show more…
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