6. The exponential distribution is often used to model...

6. The exponential distribution is often used to model waiting times. For example, at a particular restaurant, the amount of time (in hours) it takes for a person to be seated (the waiting time) is an exponential random variable with parameter ? = 4. a. Write out a formula for the p.d.f. for the waiting time X. b. What is the mean waiting time? What is the variance of the waiting times? (You may use the result of the previous exercise to compute these rather than computing the integrals.) c. What is the probability that the waiting time will be less than 15 minutes? d. What is the probability that the waiting time will be more than 30 minutes?

Transcript

00:01 Hello students in this question, we have given the parameter of the exponential random variable, which is lambda is equal to 4.

00:08 That means we have given that the random variable x follows the exponential distribution with parameter lambda is equal to 4.

00:16 Now by using this in the first question, we have to write the formula for the pdf that is probability density function for the waiting time x.

00:26 So now x is the random variable which follows the exponential then the pdf for the waiting time is f of x is equal to lambda into e raised to minus lambda into x and now this is equal to 4 into e raised to minus 4 into x.

00:55 So this is the pdf for the exponential distribution.

00:57 Then in the next question, we have to calculate the mean waiting time and also variance of the waiting time.

01:04 So now we know that the mean waiting time of the exponential distribution mean waiting time is equal to 1 divided by lambda that is equal to 1 divided by 4 is equal to 0 .25.

01:22 Then the variance is 1 divided by lambda square, which is equal to 1 divided by 16 and then in the next question, we have to calculate the probability that the waiting time will be less than 15 minutes.

01:39 So probability of x is less than 15.

01:43 Now, we know the cdf that is the cumulative distribution function of the exponential.

01:50 So cdf of the exponential distribution is capital f of x that is equal to probability of x is less than or equal to x is equal to 1 minus e raised to minus lambda into x...

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