# Thread: Probability functions

1. ## Probability functions

The life-time, t, of a bulb in a traffic signal is a random variable with density

f(t)=1 for (1<=t<=2)
f(t)=0 otherwise

where t is measure in yrs. What is the probability as a function of y that the bulb fails in less than y yrs? The traffic signal contains 3 bulbs. Assuming they fail independently, what is the probability as a function of z that none of the bulbs have to be replaced in z years?

I don't really understand what f(t) is, so I'm having problems doing this question. Would I be right to say that the last bit should be done using Poisson distribution? Help would be appreciated.

2. Originally Posted by free_to_fly
The life-time, t, of a bulb in a traffic signal is a random variable with density

f(t)=1 for (1<=t<=2)
f(t)=0 otherwise

where t is measure in yrs. What is the probability as a function of y that the bulb fails in less than y yrs? The traffic signal contains 3 bulbs. Assuming they fail independently, what is the probability as a function of z that none of the bulbs have to be replaced in z years?

I don't really understand what f(t) is, so I'm having problems doing this question. Would I be right to say that the last bit should be done using Poisson distribution? Help would be appreciated.
Let T be the random variable lifetime of a bulb.

(a) $\displaystyle \Pr(T < y) = \int_{-\infty}^{+\infty} f(t) \, dt = \int_1^y 1 \, dt = y - 1$.

(b) $\displaystyle \Pr(T > z) = \int_{-\infty}^{+\infty} f(t) \, dt = \int_z^2 1 \, dt = 2 - z$.

Let X be the random variable number of bulbs that last more than z years .

X ~ Binomial(n = 3, p = 2 - z)

Calculate Pr(X = 3).