Estimating number of failures

Question

I have a set ${X}$ of $N$ Weibull-distributed variables that are differently parameterized. Let's denote as $S_i(t)$ the probability of survival for variable $i$ at time $t$. $F_i(t) = 1 - S_i(t)$ is the failure probability for variable $i$. I'd like to estimate the expected # of total failures at time $T$. Intuitively, since the ${X}$ are independent, the probability of all ${X}$ surviving until $T$ is $\Pi_i S_i(T)$, and the probability of 1 failure in the set is then $F_{tot} = 1 - \Pi_i S_i(T)$, and the expected # of failures is then $F_{tot} * N$. But I'm not sure if my logic is quite correct. If not, what's the right approach to calculating total # of failures at a given time?

Answer 1

Let us look at $x_1$, without loss of generality. The probability that it fails before time $T$ is $F_1(T)$. This is the expected value of a variable that takes on the value $1$ if $x_1 < T$ and $0$ otherwise:

$$F_1(T) = \int_0^{\infty}1(x < T)f(x)dx$$

Similarly, all the other $F_i(T)$ are expected values of the appropriate indicator variable. This motivates a view of the $F_i(T)$ as expected values - the $F_i(T)$ can be interpreted as the expected number of failures of a single observation of variable $i$ before time $T$, which is just the probability of failure before time $T$.

Addition based on DeltaIV's comment:

Now, the total number of failures $n(T)$ before time $T$ can be written:

$$n(T) = \sum_{i=1}^NX_i(T)$$

where, for each fixed $T$, $X_i(T)$ is either 1 or 0 depending upon whether a failure occurred or not respectively. You can see how the values 1 or 0 match up with the indicator function values in the first equation. This is just the sum of $N$ Bernoulli variates with (possibly) different probabilities, which probabilities equal the $F_i(T)$.

The total expected number of failures before time $T$, $\mathbb{E}X(T)$, is therefore just the sum of the expected number of failures for each of the different items:

$$\mathbb{E}X(T) = \sum_i F_i(T)$$