Question

Let X be the number of life years before certain kind of CFL bulb needs replacement . Let X have the probability mass function is defined as

f(x)=alphax**3,x=0,1,2,3,4

Where alpha is constant

(a) Find the values of constant alpha

(b) find the cumulative distribution function F to X

Answer 1

(a)

$$\alpha(0)^3+\alpha(1)^3+\alpha(2)^3+\alpha(3)^3+\alpha(4)^3=1$$

$$\alpha=0.01$$

(b)

$$f(x)=\begin{cases} 0 &x=0\\ 0.01&x=1 \\ 0.08 &x=2 \\ 0.27 & x=3 \\ 0.64 & x=4\\ \end{cases}$$

Then

$$F(x)=\begin{cases} 0 &x<1\\ 0.01&1\le x<2 \\ 0.09 &2\le x<3 \\ 0.36 & 3\le x<4 \\ 1 & x\ge 4 \\ \end{cases}$$