Question

How to show that all the probability adds to 1

2112015S1 Exam Q3] A newly designed super-bright LED automobile head-light set exhibits varying time till failure (TTF). Tom, the marketing analyst, obtained the following probability distribution for the LED head-light set's TTF from the quality control engineer: P(X:: x)-me-2) χ>2.xis integer [1] x! (e 2) where X is the random variable of number of years from initial usage until failure occurs, and x is a constant integer no smaller than 2. Assume that the LED head-light sets have sufficient built-in quality that they will never fail in the first two years of usage, and subsequently will fail only at the end of discrete number of years (a) Show that the formula given in [1] is indeed a probability distribution. 4 marks) Sourced from NEA website http://www.nea.gov.sg/weather-climate/elimate/weather-statistics, accessed on 2016 Oct 02. 2018SEM2 Tutorial 2 Page1

Answer 1

The formula given in [1} is indeed a probability distribution if th Total Probability = 1

i.e.,

To show:

(1)

Proof:
LHS of(1) is written as follows:

(2)

By Power Series, we have:

$e=1+rac{1}{1!}+rac{1}{2!}+...infty$ (3)

From (3), we get:

i.e.,

(4)

Substituting (4), equation (1) becomes:

$x(e-1)=1$

This is the required result.