Question

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.

Answer 1

The cumulative distribution function (CDF) for X is constructed by summing the probabilities up to and including each value of X. The CDF values for X=0,1,2,3,4 are 0.41, 0.78, 0.94, 0.99, and 1.00, respectively.

Step-by-step explanation:

To construct the cumulative distribution function (CDF) of X, we need to sum the probabilities f(x) up to a given value of x. This provides the probability that the random variable X is less than or equal to that value. Let's denote the CDF by F(x).

• For x=0: F(0)=P(X≤0)=0.41
• For x=1: F(1)=P(X≤1)=P(X=0)+P(X=1)=0.41+0.37=0.78
• For x=2: F(2)=P(X≤2)=F(1)+P(X=2)=0.78+0.16=0.94
• For x=3: F(3)=P(X≤3)=F(2)+P(X=3)=0.94+0.05=0.99
• For x=4: F(4)=P(X≤4)=F(3)+P(X=4)=0.99+0.01=1.00

Thereby, the cumulative distribution function F(x) for the discrete random variable X is as follows:

• F(0)=0.41
• F(1)=0.78
• F(2)=0.94
• F(3)=0.99
• F(4)=1.00

Answer 2

Step-by-step explanation:

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!