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# 數學│機率論問題│請幫幫忙│20點│感恩感謝

Q：A random sample X1,X2,.....Xn of size n is taken from a Poisson distribution with a mean of λ,0<λ<∞.

(a) Show that the maximum likelihood estimator for λ is X^.

(b) Let X equal the number of flaws per 100 feet of a used computer tape.Assume that X has a Poisson distribution with a mean of λ.If 40 observations of X yielded 5 zeros ,7 ones,9 threes,5 fours,1 five,and 1 six,find the maximum likelihood estimate ofλ.

P.S：x^= ∧

X

Ans：(b)mean=89/40=2.225

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最大概似估計（maximum likelihood estimation）是一種統計方法，它用來求一個樣本集的相關機率密度函數的參數。

Poisson distribution 的機率密度函數：

P（X=k）=

概似函數（likelihood function）為

兩邊取自然對數：

然後對參數λ微分，並令之為零：

就可以得到：

Let X equal the number of flaws per 100 feet of a used computer tape.Assume that X has a Poisson distribution with a mean of λ.If 40 observations of X yielded 5 zeros ,7 ones, 12 twos, 9 threes,5 fours,1 five,and 1 six,find the maximum likelihood estimate ofλ. （原PO題目好像漏掉12個2。）

So, the maximum likelihood estimate ofλ= [ (7*1)+(12*2)+(9*3)+(5*4)+(1*5)+(1*6) ] / 40 = 89/40 = 2.225。