1. ## Distribution help

The question:

$p(X = k) = \frac{c}{k!}$, k = 0, 1, 2, ...

a) Determine the value of c.
b) Calculate $P(X = 2)$
c) Calculate $P(X < 2)$
d) Calculate $P(X \ge 4)$

I think I know how to do b), c) and d), but a) is giving me trouble.

I tried this:

$\sum\limits_{i = 0}^{k} \frac{c}{i!} = 1$ (since it's a distribution)

$\sum\limits_{i = 0}^{k} \frac{1}{i!} = \frac{1}{c}$

I'm confident I've done this completely wrong. Any advice?

2. Originally Posted by Glitch
The question:

$p(X = k) = \frac{c}{k!}$, k = 0, 1, 2, ...

a) Determine the value of c.
b) Calculate $P(X = 2)$
c) Calculate $P(X < 2)$
d) Calculate $P(X \ge 4)$

I think I know how to do b), c) and d), but a) is giving me trouble.

I tried this:

$\sum\limits_{i = 0}^{k} \frac{c}{i!} = 1$ (since it's a distribution)

$\sum\limits_{i = 0}^{k} \frac{1}{i!} = \frac{1}{c}$

I'm confident I've done this completely wrong. Any advice?
You need to use the standard power series $\displaystyle e^x = \sum_{i=0}^{+\infty}\frac{x^i}{i!}$.

3. Ahh, so 'x' in the power series is just '1', which makes c = 1/e?

4. Originally Posted by Glitch
The question:

$p(X = k) = \frac{c}{k!}$, k = 0, 1, 2, ...

a) Determine the value of c.
b) Calculate $P(X = 2)$
c) Calculate $P(X < 2)$
d) Calculate $P(X \ge 4)$

I think I know how to do b), c) and d), but a) is giving me trouble.

I tried this:

$\sum\limits_{i = 0}^{k} \frac{c}{i!} = 1$ (since it's a distribution)

$\sum\limits_{i = 0}^{k} \frac{1}{i!} = \frac{1}{c}$

I'm confident I've done this completely wrong. Any advice?
... that isn't completely wrong... only the condition for c is...

$\displaystyle \sum_{k=0}^{\infty} \frac{1}{k!} = \frac{1}{c}$ (1)

... i.e. the sum has infinite terms... Once You have c the other questions are relatively easy, for example...

$\displaystyle P (x<2) = c\ (1+1)= 2\ c$

Kind regards

$\chi$ $\sigma$