In terms of the sample mean x n 1 n n i 1 x i the

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in terms of the sample mean ¯ X n = 1 n n i =1 X i . The subscript n in ¯ X n is there to remind us that we have the average of n values. It doesn’t refer to the n th sampled value X n . 3
b) This part is a simulation study in which you already know λ and are examining how close your estimate comes to this known value. Take λ = 0 . 25 . We have assigned it the name lam because lambda means something else in Python. Repeat the following simulation 10,000 times. - Generate a sample of 400 i.i.d. exponential ( lam ) variables. Use stats.expon.rvs(scale = 1/lam, size=400) . - Calculate your MLE ˆ λ 400 based on this sample. Then draw the histogram and find the mean of your 10,000 MLEs.
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[6]: # The mean of your 10000 mle's np . mean(mle_400) [6]: 0.2506214495007068 c) Use facts about sums and linear transformations to find the distribution of ¯ X n with little or no calculation. Recognize it as one of the famous ones and provide its name and parameters. Use it to find E ( ˆ λ n ) . d) Is ˆ λ n an unbiased estimate of λ ? If it is biased, does it overestimate on average, or does it underestimate? Is it asymptotically unbiased? That is, does E ( ˆ λ n ) converge to λ as n → ∞ ? e)

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