# Thread: solve it in Poisson Theorem

1. ## solve it in Poisson Theorem

look the question ,i could'nt solve it perfectly
my question is :how it from step 1 to step 2.and how it from step 2 to step 3?
thx

2. Originally Posted by neworld222
look the question ,i could'nt solve it perfectly
I don't see a question here. Could you give some mone expalantion.

The only thing here that I see that could need explanation is that:

$
e^x = \lim_{n \to \infty} \left(1+\frac{x}{n} \right)^n
$

so as $100$ is large (for our purposes):

$
\left(1-\frac{1}{500}\right)^{100} \approx e^{-1/5}
$

RonL

3. Originally Posted by CaptainBlack
I don't see a question here. Could you give some mone expalantion.

The only thing here that I see that could need explanation is that:

$
e^x = \lim_{n \to \infty} \left(1+\frac{x}{n} \right)^n
$

so as $100$ is large (for our purposes):

$
\left(1-\frac{1}{500}\right)^{100} \approx e^{-1/5}
$

RonL
i'd update my question,please view it again

4. Originally Posted by neworld222
look the question ,i could'nt solve it perfectly
my question is :how it from step 1 to step 2.and how it from step 2 to step 3?
thx
I don't understand what exactly your question is about how to go from step 2 to step 3? Just plug it into a calculator.

-Dan

5. Originally Posted by topsquark
I don't understand what exactly your question is about how to go from step 2 to step 3? Just plug it into a calculator.

-Dan
i get subject on a book.
there says it go from step 2 to setp 3 by 'poisson theorem'.
i don't comprehend about 'poisson theorem'.so i need help in this math forum

6. Poisson's Theorem says:
$\frac{n!}{k!(n - k)!}p^kq^{n-k} \approx e^{-np} \frac{(np)^k}{k!}$
so this would refer to going from step 1 to step 2, not 2 to 3.

But I don't see the applicability here. The only sensible (to me) way to approximate something like
$\left ( 1 - \frac{1}{500} \right )^{100}$
is to use the binomial approximation:
$\left ( 1 - \frac{1}{500} \right )^{100} \approx 1 - 100 \cdot 1 \cdot 1^{99} \cdot \left ( \frac{1}{500} \right ) ^{1}$

We could use Poisson's theorem on the second term and get that
$100 \cdot \left ( \frac{1}{500} \right ) ^1 \cdot 1^{99} \approx e^{-100 \cdot \frac{1}{500}} \frac{(100 \cdot \frac{1}{500})^1}{1!} \approx 0.163746$
but this is a horrid approximation.

-Dan

7. Originally Posted by neworld222
look the question ,i could'nt solve it perfectly
my question is :how it from step 1 to step 2.and how it from step 2 to step 3?
thx

There is no use for Poisson's theorem here. I have already explained in
another post how you get to line 2 from line 1.

Line 3 is obtained by evaluating line 2 on your calculator.

Poisson's theorem tell you how to approximate a binomial distribution (under
certain conditins) by a Poisson distribution, and if that is involved in this
question it occured before your line 1.

RonL

8. i see.if there used Poisson theory.it should be between step 1 and step 2.not 2 and 3.
both CaptainBlack's and topsquark methods can solve this subject(lim. and Poisson theory).
i thought they don't typeset my book's clearly,and i did'nt see the subject carefulness enough,so i thought the Poisson Theory used in the 3rd step
now i have understand how y works to step 3.
thx all of you.