PROBABILITY 07
Discrete Distributions
A discrete probability distribution assigns probability to each possible value of a discrete random variable.
Probability mass function
For a discrete random variable \(X\), define
\[\boxed{p_X(x)=P(X=x)}\]
This is the probability mass function, or pmf.
1. Conditions
A valid pmf satisfies
\[p_X(x)\ge0\]and
\[\boxed{\sum_x p_X(x)=1}.\]2. Example
If \(X\) is the number of heads in two fair coin tosses,
\[P(X=0)=\frac14,\qquad P(X=1)=\frac12,\qquad P(X=2)=\frac14.\]3. Cumulative distribution function
The cumulative distribution function is
\[\boxed{F_X(x)=P(X\le x)}.\]For a discrete variable it increases in jumps as values with positive probability are passed.
4. Bernoulli distribution
A Bernoulli variable represents one success/failure trial:
\[P(X=1)=p,\qquad P(X=0)=1-p.\]5. Binomial distribution
If \(X\) counts successes in \(n\) independent Bernoulli trials,
\[P(X=k)=\binom nk p^k(1-p)^{n-k}.\]6. Poisson distribution
A Poisson variable is often used for event counts:
\[P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!},\qquad k=0,1,2,\ldots\]Poisson distributions become especially important in continuous-time stochastic processes.
7. Central idea
\[\boxed{\text{discrete value}\rightarrow\text{probability mass}}\]