Waiting times to receive food after placing an order at the sandwich shop follow an exponential distribution with a mean of 46 seconds.?

Calculate the probability a customer waits:

a.) Less than 26 seconds
b.) More than 105 seconds
c.) Between 38 and 60 seconds
d.) Fifty percent of the patrons wait less than how many seconds? What is the median?

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To solve these probability questions related to exponential distribution, we can use the cumulative distribution function (CDF) of the exponential distribution. The exponential CDF is given by:

CDF(x; λ) = 1 - exp(-λx)

where x is the random variable, and λ is the rate parameter.

Given the mean of the distribution is 46 seconds, we can calculate the rate parameter λ using the formula:

λ = 1 / mean

Let's calculate the rate parameter λ first:
λ = 1 / 46 = 0.02174

a.) To calculate the probability that a customer waits less than 26 seconds, we can substitute the values into the exponential CDF. The formula becomes:

CDF(26; 0.02174) = 1 - exp(-0.02174 * 26)

By evaluating the expression, you'll find the probability that a customer waits less than 26 seconds.

b.) To calculate the probability that a customer waits more than 105 seconds, we can use the complement rule. The probability of waiting more than 105 seconds is equal to 1 minus the probability of waiting less than or equal to 105 seconds:

P(x > 105) = 1 - CDF(105; 0.02174)

Substituting the values and evaluating the expression will give you the probability.

c.) To calculate the probability that a customer waits between 38 and 60 seconds, we can subtract the CDF values at these two points:

P(38 < x < 60) = CDF(60; 0.02174) - CDF(38; 0.02174)

d.) To find the time at which 50% of the patrons wait less than, we can use the quantile function (inverse of the CDF) of the exponential distribution. The formula for the quantile function is:

quantile(p; λ) = -log(1 - p) / λ

Let's calculate the time at which 50% of the patrons wait less than by substituting p=0.5 and the rate parameter:

quantile(0.5; 0.02174) = -log(1 - 0.5) / 0.02174

The result will give you the time at which 50% of the patrons wait less than, which is also the median of the distribution.