Question

1- What is the Probability Mass Functions & how to calculate and give two different example?

2- What is Poisson Distribution and how to calculate and give two different example?

Answer 1

1) Let X be a discrete random variable with range RX={x1,x2,x3,...} (finite or countably infinite). The function

P_X(x_k) =P(X=xk), for k=1,2,3,...,

is called the probability mass function (PMF) of X if it satisfies two conditions

1) p(X=Xk) ≥ 0

2) P(X=xk) =1

e.g.

I toss a fair coin twice, and let X be defined as the number of heads I observe. then range of X, Rx, as well as its probability mass function Px is as follows.

• Here, our sample space is given by

  S={HH,HT,TH,TT}.

  The number of heads will be 0, 1 or 2. Thus

  RX={0,1,2}.

  Since this is a finite (and thus a countable) set, the random variable X is a discrete random variable. Next, we need to find PMF of X. The PMF is defined as

  PX(k)=P(X=k) for k=0,1,2.

  We have

  PX(0)=P(X=0)=P(TT)=1/4,

  PX(1)=P(X=1)=P({HT,TH})=1/4+1/4=1/2,

  PX(2)=P(X=2)=P(HH)=1/4.

.