Question

The amount of time that an electrician requires to switch on the generator in a theater when the power goes off is a random variable having an exponential distribution with a mean of 4 minutes. What is the Probability that the electrician requires less than 2 minutes on atleast 4 of the next 6 days to switch on the generator.

Answer 1

Answer on Question #53412 – Math – Statistics and Probability

Question

The amount of time that an electrician requires to switch on the generator in a theater when the power goes off is a random variable having an exponential distribution with a mean of 4 minutes. What is the Probability that the electrician requires less than 2 minutes on at least 4 of the next 6 days to switch on the generator.

Solution

If the mean of a random variable having an exponential distribution is $\mu = 4$, then $\lambda = \frac{1}{\mu} = \frac{1}{4}$.

If $X$ is a random variable having an exponential distribution, then

$$p = P(X < 2) = F(2) = 1 - e^{-\frac{1}{4} \cdot 2} = 1 - e^{-1/2} = 0.393469$$

is probability that the electrician requires less than 2 minutes in one day, $q = 1 - p$, $F$ is the cumulative distribution function of a random variable having an exponential distribution.

Using binomial distribution, probability that the electrician requires less than 2 minutes on at least 4 of the next 6 days to switch on the generator is

$$P(Z \geq 4) = P(Z = 4) = P(Z = 5) + P(Z = 6) = \binom{6}{4} p^4 q^2 + \binom{6}{5} p^5 q + \binom{6}{6} p^5 q = 0.170295,$$

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.

Answer: 0,170295.

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