Question

The distribution function of a discrete random variable X is given by

F_x(k) = $\begin{Bmatrix} 0, & k < 1 \\ 0.1, & 1 \leq k <2 \\ 0.3, & 2 \leq k <3\\ 0.7, & 3 \leq k <4 \\ 0.8 & 4 \leq k < 5\\ 1 & k \geq 5 \end{Bmatrix}$

1) What are the possible values of X?

2) Find the probability mass function of X.

Answer 1

1) The possible values of $X$ are $1,2,3,4,5$

2)

$$F(1)=P(X\le1)=p(1)=0.1$$

$$F(2)=P(X\le2)=p(1)+p(2)=0.3$$

$$p(2)=0.3-0.1=0.2$$

$$F(3)=P(X\le3)=p(1)+p(2)+p(3)=0.7$$

$$p(3)=0.7-0.3=0.4$$

$$F(4)=P(X\le4)=p(1)+p(2)+p(3)$$

$$+p(4)=0.8$$

$$p(4)=0.8-0.7=0.1$$

$$F(5)=P(X\le5)=p(1)+p(2)+p(3)$$

$$+p(4)+p(5)=1$$

$$p(5)=1-0.8=0.2$$

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 0.1 & 0.2 & 0.4 & 0.1 & 0.2 \\ \end{array}$$