In a book of $500$ pages, there are $1000$ typographical errors. Assume that errors are equally likely to be on any page (that is, the page number of each error follows the uniform distribution), independently of each other. Let $X$ be the number of errors on the last page. State the distribution which $X$ follows, the expectation $E[X]$ and the exact formula for $P(X ≤ 3)$. Estimate $P(X ≤ 3)$ using the Poisson approximation.
Solution Attempt:
Distribution of X: Binomial, with $X$~$(1000, \frac{1}{500})$
Expectation of X:
Exact $P(X ≤ 3)$:
Poisson approximation of $P(X ≤ 3)$: We must have $\lambda = 2$ since $\frac{1000}{500}=2$. $P(X ≤ 3)$ = $\sum_{k=0}^{3}{\frac{e^{-2} \cdot 2^k} {k!}} = 0.8571 $
I want to check whether the distribution type is correct before continuing.
Yes, that is correct. Each of the $1000$ errors can be conceptualized as an independent Bernoulli trial, whose probability of occurring on the last page is $p = 1/500$. Therefore, the total number of errors on the last page is a binomial random variable with $n = 1000$ such trials.