The lifetimes of batteries are independent exponential random variables, each having parameter λ. A flashlight needs 4 batteries to work. If one has a flashlight and a stockpile of n batteries, what is the expected time that the flashlight can operate?
To find the expected time that the flashlight can operate, we need to calculate the expected value of the minimum lifetime among the four batteries.
Let's denote the minimum lifetime among the four batteries as X. Since the lifetimes of batteries are independent exponential random variables with parameter λ, the probability density function (PDF) of each individual battery is given by:
f(x) = λ * e^(-λx) for x ≥ 0
The cumulative distribution function (CDF) of each individual battery is given by:
F(x) = 1 - e^(-λx) for x ≥ 0
Now, we know that the minimum lifetime among the four batteries (X) is given by:
X = min(B1, B2, B3, B4)
To find the distribution function of X, we can use the fact that the minimum of exponential random variables follows an exponential distribution with the sum of the parameters λ:
F_X(x) = P(X ≤ x) = 1 - P(X > x) = 1 - P(B1 > x, B2 > x, B3 > x, B4 > x)
Since the lifetimes of the batteries are independent, we can use the fact that the probability of the intersection of independent events is equal to the product of the individual probabilities:
F_X(x) = 1 - P(B1 > x) * P(B2 > x) * P(B3 > x) * P(B4 > x)
Using the cumulative distribution function for each battery:
F_X(x) = 1 - (1 - F(x))^4
Differentiating both sides of the equation with respect to x, we can find the probability density function of X:
f_X(x) = d/dx [F_X(x)] = 4 * (1 - F(x))^3 * f(x)
Since f_X(x) is the probability density function of the minimum lifetime, we can calculate the expected value (E[X]) as follows:
E[X] = ∫[0, ∞] x * f_X(x) dx
Substituting the expression for f_X(x) derived earlier:
E[X] = ∫[0, ∞] x * 4 * (1 - F(x))^3 * f(x) dx
Simplifying this integral will give us the expected time that the flashlight can operate. However, since this is a complex mathematical calculation, it may be easier to approximate the expected value using numerical methods or software.