Question

The number of months that a light bulb is functioning is exponentially distributed with parameter λ = 1/10 .

What is the probability that a light bulb will burn for at least 4 months?

Find the mean and the standard deviation.

What is the probability that a light bulb will burn for at least 12 months, given that it has burned for 4 months?

Answer 1

Let X denote the random variable representing the number of months that a light bulb is functioning. Now, we are given that X is exponentially distributed with parameter λ = 1/10.

=> X ~ Exponential(λ = 1/10)

Thus, the probability density function of X is given by:

Moreover, the cumulative distribution function of X is given by:

Part 1

The probability that a light bulb will burn for at least 4 months is given by:

Part 2

Before we find the mean and the variance of X, we note that if Y~Exp(r), then the mean and standard deviation of Y are given by:

E(Y) = S.D.(Y) = 1/r

Thus, the mean of X is given by:

Moreover, the standard deviation of X is given by:

Part 3

The probability that a light bulb will burn for at least 12 months, given that it has burned for 4 months is given by:

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