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 0 1 2 3 4

f (x) 0.41 0.37 0.16 0.05 0.01

Construct the cumulative distribution function of X.

Hello!

X: number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width.

X: 0; 1; 2; 3; 4

f (x) : 0.41; 0.37; 0.16; 0.05; 0.01

The cumulative distribution function is defined as the probability of X being equal or less to a certain value x₀, symbolically: F (X) = P (X≤x₀)

and you can calculate it by adding the absolute frequencies of each value of X.

For X=0

f (0) = 0.41

F (0) = P (X≤0) = f (0) = 0.41

⇒ In this case, since zero is the first value of the variable, meaning there are no values of probability below it, the absolute frequency and the cumulated frequency will be the same.

For X=1

f (1) = 0.37

F (1) = P (X≤1) = f (0) + f (1) = 0.41 + 0.37 = 0.78

For X=2

f (2) = 0.16

F (2) = P (X≤2) = f (0) + f (1) + f (2) = 0.78 + 0.16 = 0.94

For X=3

f (3) = 0.05

F (3) = P (X≤3) = f (0) + f (1) + f (2) + f (3) = 0.94 + 0.05 = 0.99

For X=4

f (4) = 0.01

F (4) = P (X≤4) = f (0) + f (1) + f (2) + f (3) + f (4) = 0.99 + 0.01 = 1

X: 0; 1; 2; 3; 4

f (x) : 0.41; 0.37; 0.16; 0.05; 0.01

F (X) : 0.41; 0.78; 0.94; 0.99; 1

Have a nice day!