Knitting machines at a factory making elastic use a laser to detect broken threa

Question

Knitting machines at a factory making elastic use a laser to detect broken threads. When a thread breaks, the machine must be stopped and the broken thread must be found and repaired by technician. Assume that the density for X, the number of times per day that a specific machine is stopped, is given by

f(x) =(16/31)(1/2)^x, x = 0,1,2,3,4

a. Find the density table for X, and verify that the sum of the probabilities given in the table is 1.

b. If x<0, what is the numerical value of F(x)?

c. If x>4, what is the numerical value of F(x)?

Explanation / Answer

a) if x = 0 then f(x) = 16/31 if x = 1 then f(x) = 8/31 if x=2 then f(x)= 4/31 if x= 3 then f(x) = 2/31

if x = 4 then f(x) = 1/31

Now sum all. we got 16/31 + 8/31 + 4/31 + 2/31 + 1/31 = 31/31 = 1

b) if x<0 then F(x) = 0 as there is no proibability assigned below x = 0

c) if x>4 then F(x) = 1 as all the probabilities assigned below x = 4

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