The exponential distribution is defined as: f(x|λ) = λ * exp(-λx) for x = 0, 0 otherwise Where λ is the rate parameter. The cumulative distribution function, which gives the probability that a random variable X following an exponential distribution is less than or equal to x, is given by: F(x|λ) = 1 - exp(-λx) Given λ = 3 (for lunch hour) and λ = 1 (for dinner time), we can calculate the probabilities as follows: a. P(X

Question

The exponential distribution is defined as:

f(x|λ) = λ * exp(-λx) for x >= 0, 0 otherwise

Where λ is the rate parameter. The cumulative distribution function, which gives the probability that a random variable X following an exponential distribution is less than or equal to x, is given by:

F(x|λ) = 1 - exp(-λx)

Given λ = 3 (for lunch hour) and λ = 1 (for dinner time), we can calculate the probabilities as follows:

a. P(X < 2) during lunch hour = F(2|3) = 1 - exp(-3*2) = 1 - exp(-6) = 1 - 0.0025 = 0.9975

b. P(X < 4) during lunch hour = F(4|3) = 1 - exp(-3*4) = 1 - exp(-12) = 1 - 0.000006 = 0.999994

c. P(X < 2) during dinner time = F(2|1) = 1 - exp(-1*2) = 1 - exp(-2) = 1 - 0.1353 = 0.8647
   P(X < 4) during dinner time = F(4|1) = 1 - exp(-1*4) = 1 - exp(-4) = 1 - 0.0183 = 0.9817

So, the answers are:

a. 0.9975
b. 0.9999
c. 0.8647 (for 2 minutes) and 0.9817 (for 4 minutes)

Please note that these are rounded to four decimal places.

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