poisson distribution

https://m.media-amazon.com/images/M/MV5BZWUzYTdhYjEtZjY2Mi00NTU4LWJmZWUtOWYwMDkyMjk5Yzc2XkEyXkFqcGdeQXVyMjg0MTI5NzQ@._V1_UY1200_CR135,0,630,1200_AL_.jpg

The taylor expansion of $ e^x

https://gyazo.com/10a59ea7fc8112964fc85d25f678dd5d

Kolmogorov's Axiom II

https://gyazo.com/475bdf92f4e22d1caa4127caef66f7e4

An unidentified destribution (not yet known what it could model)

We thought of using this because the series converges to a constant.

https://gyazo.com/d1617ef08efa1c7651371684da181ddb

Substitute our distribution into Axiom II

https://gyazo.com/fc0764d4ca50e1fc78605969a00e0ef3

We find C to be:

https://gyazo.com/c3a3692a15bc9de2ba99bcc4410c8f82

Then, we substitute C into our original distribution.

https://gyazo.com/2f7209b9993c5db2261e46bb17f891f5

Below is the Taylor expansion of $ e^{\lambda t}

https://gyazo.com/50e1505bbaa7d03985abb1b52a5945b5

When the same principles are applied to the initial distribution where $ \lambda is substituted with $ \lambda t, below is obtained:

https://gyazo.com/241d5be9b00e608b61d930d2fede7a9d

The basis of this distribution is that $ 0\leq k \leq \infty

$ \Delta t is then retroactively added in

This P(=k) models the probability of k occurances, and with $ \Delta t added in, it specifies the timeframe that the occurances are being counted on.https://gyazo.com/eb37958525cdd78e05860a37193c929e

https://gyazo.com/2f52ebb727494888deb1a7c0e16b67ac