The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given by x 01234 f(x) 0.41 0.37 0.16 0.05 0.01 Construct the cumulative distribution function of X.

$F(0)=P(X\le 0)=P(X=0)=0.41$

$F(1)=P(X\le1)=P(X=0\ or\ X=1)$

$=P(X=0)+P(X=1)=0.41+0.37=0.78$

$F(2)=P(X\le2)=P(X=0\ or\ X=1\ or\ X=2)$

$=P(X=0)+P(X=1)+P(X=2)$

$=0.78+0.16=0.94$

$F(3)=P(X\le3)=P(X=0)+P(X=1)$

$+P(X=2)+P(X=3)=0.94+0.05=0.99$

$F(4)=P(X\le4)=1$

The cumulative distribution of $X$ is

$F(x) = \begin{cases} 0 &x<0 \\ 0.41 & 0\le x<1 \\ 0.78 & 1\le x<2 \\ 0.94 & 2\le x<3 \\ 0.99 & 3\le x<4 \\ 1.00 & x\ge 1 \end{cases}$