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

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(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

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

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