Discrete Probability Distributions

E(X)=\sum xp(x)

V(X)=E[(X-\mu)^2]=\sum (x-\mu)^2p(x)=E(X^2)-\mu^2

E(X^2)=\sum x^2p(x)

Binomial Probability Distribution

Only two outcomes – fixed number of trials – independent events

p(x)={n \choose x}p^xq^{n-x}, x=0,1,2,\ldots ,n

Where: p is the probability of success, q=1-p is the probability of failure and n is the number of trials.

\mu=E(X)=np

\sigma^2=V(X)=npq=np(1-p) and E(X^2)-E(X)

M^{(t)}_x=[(1-p)+pe^t]^n

Bernoulli Probability Distribution

Only two outcomes 0 and 1.

p(x)=p^xq^{1-x},

Where: p is the probability of success, q=1-p is the probability of failure.

\mu=E(X)=p

\sigma^2=V(X)=pq=p(1-p)

M^{(t)}_x=q+pe^t

Geometric Probability Distribution

Number of trials until FIRTS SUCCESS.

p(x)=q^{x-1}p=(1-p)^{x-1}p

Where: p is the probability of success and q=1-p is the probability of failure.

\mu=E(X)=\frac{1}{p}

\sigma^2=V(X)=\frac{1-p}{p^2}

M^{(t)}_x=\frac{pe^t}{1-(1-p)e^t}

Negative Binomial Probability Distribution

How many trials for r successes. I.e. number of trials where 2nd, 3rd etc. success will occur.

p(x)={x-1 \choose r-1}p^rq^{x-r}, x=r,r+1,r+2,\ldots

Where: p is the probability of success, q=1-p is the probability of failure and r is the numer of successes.

\mu=E(X)=\frac{r}{p}

\sigma^2=V(X)=\frac{r(1-p)}{p^2}

M^{(t)}_x=(\frac{pe^t}{1-qe^t})^r

Hypergeometric Probability Distribution

Conditional probability of a selection. Thus trials are not independent.

E.g. Bag with 4 red and 7 green tokens, thus n=11. Randomly select m=3 without replacement. Number of red tokens =r. Probability of selecting 1 red:

p(x)=\frac{{r \choose x}{n-r \choose m-x}}{{n \choose m}}

\mu=E(X)=\frac{mr}{n}

\sigma^2=V(X)=n(\frac{r}{n})(\frac{n-r}{n})(\frac{n-m}{n-1})

Poisson Probability Distribution

Number of successes in a time interval.

p(x)=\frac{\lambda^x}{x!}e^{-\lambda}

\lambda is the mean number of successes per interval. \lambda=np.

\mu=E(X)=\lambda

\sigma^2=V(X)=\lambda

M^{(t)}_x=e^{\lamda e^t-\lambda}

The Poisson probability distribution can be used to approximate the binomial probability distribution for large n and small p.