# Probability distribution function

1. Apr 4, 2015

### SMC

1. The problem statement, all variables and given/known data
The technical specification of a particular electrical product states that the probability of its failure with time is given by the function:

f(t) = 1 - ke^(-t/t0) if 0 < t < tmax
f(t) = 0 if t > tmax

where t is the time of service in years, and the constant t0 = 100 years defines the characteristic deterioration time. If the maximal life span of the product tmax is 10 years,
find the value of constant k , explaining its meaning,

2. Relevant equations
i thought i was supposed to find the integral of 1 - ke^(-t/t0) with respect to t between 0 and 10 years and put the answer equal to 1. then find k from that.

3. The attempt at a solution

now this is a past exam paper question so i actually have the partial solution to the question which says that:

k = -9/(t0(e^(-10/t0) - 1))

but what i get is k = -9/(t0e^(-10/t0))

I'm pretty sure i'm doing the integral correctly so there must be some step i'm missing that accounts for that extra -1 in the denominator.

Last edited: Apr 4, 2015
2. Apr 4, 2015

### Staff: Mentor

Did the past exam question have exactly the same problem statement?
The second solution looks odd. It gives wrong results for very large t0, for example. How did you get it?

Why do both solutions depend on t? They should not do that. Is that tmax?

3. Apr 4, 2015

### SMC

yeah sorry that t should be tmax = 10... i've edited it

4. Apr 4, 2015

### SMC

i just did the integral 0 and 10 of the function given and put it equal to 1 because the probability of it failing within a 10 year period is 100%. once i've done the integral I get:

10 - k*e^(-10/t0)*(-t0) = 1 (int. between 0 and 10 of 1 is 10 and int. between 0 and 10 of k*e^(-t/t0) is k*e^(-10/t0)*(-t0) )

then i find k from that. is this wrong?

5. Apr 4, 2015

### SMC

OMG I just realised what I did! i'm really sorry! never mind! please ignore!

6. Apr 4, 2015

### SMC

i considered e^0 = 0 i've made this mistake quite a few times especially in integrlas!