Question

A random variable X has pdf:

X

1

2

3

f(x)

1/8

3/8

3/8

1/8

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 0 & 1 & 2 & 3 \\ \hline f(x) & 1/8 & 3/8 & 3/8 & 1/8 \\ \end{array}$$

a)

$$F(0)=P(X\leq 0)=P(X=0)$$

$$=\dfrac{1}{8}$$

$$F(1)=P(X\leq 1)=P(X=0)+P(X=1)$$

$$=\dfrac{1}{8}+\dfrac{3}{8}=\dfrac{1}{2}$$

$$F(2)=P(X\leq2)=P(X=0)+P(X=1)+P(X=2)$$

$$=\dfrac{1}{8}+\dfrac{3}{8}+\dfrac{3}{8}=\dfrac{7}{8}$$

$$F(1)=P(X\leq 3)=P(X=0)+P(X=1)$$

$$+P(X=2)+P(X=3)=\dfrac{1}{8}+\dfrac{3}{8}+\dfrac{3}{8}+\dfrac{1}{8}=1$$

$$F(x) = \begin{cases} 0 &x<0 \\ 1/8 & 0\leq x<1\\ 1/2 & 1\leq x<2\\ 7/8 & 2\leq x<3\\ 1 & x\geq 3\\ \end{cases}$$

b)