Accidents occur in a factory at the rate of 2 per week. Assume that accidents happen randomly and independently of each other.
I'm not sure my answers are correct but I'll show my work so far:
a) What is the probability that the time to the first accident is greater than 2 weeks?
$P(X > 2) = 1 – P(X = 0) + P(X = 1)$
$1-\left(\frac{2^1}{e 1!}+\frac{2^0}{e^2 0!}\right) = .129$
b) What is the probability that the time to the first accident is less than 2 days (2/7 week)?
$P(X < 2) = P(X = 0) + P(X = 1)$
$\frac{2^1}{e 1!}+\frac{2^0}{e^2 0!}\ = .871$
I'm also asked to find the mean time to the first accident and the variance of the time to the first accident but am unsure of how to approach that.
Best Answer
HINT The time to the first accident is a continuous random variable, not a discrete one. Try to model this with an exponentially distributed random variable, with the pdf $$ f(x) = \lambda e^{-\lambda x}, x > 0. $$