The story of every distribution - Discrete Distributions

Distributions play an important role in the life of every Statistician. I coming from a non-statistic background am not so well versed in these and keep forgetting about the properties of these famous distributions. That is why I chose to write my own understanding in an intuitive way to keep a track. One of the most helpful way to learn more about these is the STAT110 course by Joe Blitzstein and his book . You can check out this Coursera course too. Hope it could be useful to someone else too. So here goes:

1. Bernoulli Distribution:

Perhaps the most simple discrete distribution of all.

Story: A Coin is tossed with probability p of heads.

PMF of Bernoulli Distribution is given by:

P(X=k)={1−ppk=0k=1P(X=k)={1−pk=0pk=1

CDF of Bernoulli Distribution is given by:

P(X≤k)=⎧⎩⎨⎪⎪01−p1k<00≤k<1k≥1P(X≤k)={0k<01−p0≤k<11k≥1

Expected Value:

E[X]=∑kP(X=k)E[X]=∑kP(X=k)

$$E[X] = 0P(X=0)+1P(X=1) = p$$

Variance:

Var[X]=E[X2]−E[X]2Var[X]=E[X2]−E[X]2

Now we find,

E[X]2=p2E[X]2=p2

and

E[X2]=∑k2P(X=k)E[X2]=∑k2P(X=k)

E[X2]…