Solving 2 Problems with Exponential RVs: CLT & Probability

[marcadams267]

I have the following two problems that I need to solve:

1. Suppose that the service time for a student enlisting during enrollment is modeled as an
exponential RV with a mean time of 1 minute. If the school expects 500 students during
enrollment period, what is the probability that the registration committee will finish enlisting all
students within a day if a day constitutes a total of 8 working hours? Use the Central Limit
Theorem.

2. Suppose that you have a class at 8:45 am and you left your house at 8:00 am. To get to class, you
ride a train, ride a taxi and then walk. The train waiting time, train trip time, taxi waiting time, taxi trip time,
and walking time are all modeled as exponential random variables with a mean time of 10 mins,
and they are assumed to be independent of each other. What is the probability that you will not
be late for class?My attempted solution:
1. Since mean time is 1 minute, and there are 480 mins in 8 hours, mean amount of students = 480
Variance = 1/λ^2, so standard deviation = 1/λ = mean = 480
Central limit theorem: Z = (S - mean)/standard_deviation --> (500-480)/480 = 0.0416666666

2. Since all the waiting times are identical, I add them up and get a mean time of 50 mins.
λ = 1/50 = 0.02
probability for an exponential random variable is λe^-(λt)
I then solve for the integral from 0 to 45 for 0.02e^-(0.02t) dt = 0.59344

Are my answers correct?

 

[StoneTemplePython]

let's tackle part 1 first

I have the following two problems that I need to solve:

1. Suppose that the service time for a student enlisting during enrollment is modeled as an
exponential RV with a mean time of 1 minute. If the school expects 500 students during
enrollment period, what is the probability that the registration committee will finish enlisting all
students within a day if a day constitutes a total of 8 working hours? Use the Central Limit
Theorem.
...

My attempted solution:
1. Since mean time is 1 minute, and there are 480 mins in 8 hours, mean amount of students = 480
Variance = 1/λ^2, so standard deviation = 1/λ = mean = 480
Central limit theorem: Z = (S - mean)/standard_deviation --> (500-480)/480 = 0.0416666666

...
Are my answers correct?

the fact that you didn't use a poisson distribution here is a red flag you need to think about. You should also explicitly write out the random variable and what is stands for... jumping straight into calculations can lead to problems like this.

As is you seem to think the problem involves adding 480 iid exponential random variables. I don't see how this can possibly be the case -- we are interested in 500 students not 48. Note: even if your setup was correct, there's a linearity problem -- the combined random variable would in fact have variance of 480 but would have standard deviation of $\sqrt{480}$ -- variance adds in the case of iid random variables but standard deviation does not-- you can't interchange sums of positive numbers and square roots due to negative convexity.

so writing this out, what you actually want to know is whether

$S_{500} \leq 480$
this is standard partial sum notation you may have seen in calculus e.g. $s_3 = x_1 + x_2 + x_3$, except we are summing random variables,

$S_{500} = X_1 + X_2 + ... + X_{500}$

where each $X_j$ represents a student, having iid arrival / service time that is an exponential random variable with parameter $\lambda = 1$

so what is
$Pr\big(S_{500} \leq 480\big) $

well this is asking whether the first 500 arrivals occur at some time less than or equal to 480 minutes, which is equivalent to asking whether the number arrivals at time 480 are at least 500 in a poisson process. i.e.

whether
$Pr\big(N(t)) \geq 500\big)$
with $t = 480$
and $N(t)$ is the 'counting' random variable which counts the number of iid exponentially distributed arrivals in $(0, t]$

and from your text you should know
$Pr\big(N(t) = k )$

is precisely given by a poisson distribution.

From here I'd suggest calculating the answer exactly in say excel, and then doing the normal approximation and comparing answers.

= = = = =
note: you didn't give any background on the course or book you are using so I had to guess on what you know. It is possible that some of what I said won't make sense. My first suggestion of completely writing out the problem and what are the random variables you're interested, and why stands in any case. Second, there are many good texts on poisson processes. These days I'd probably recommend the freely available book by Blitzstein and Hwang which has a nice chapter on Poisson processes that you may want to go through. See here:

https://projects.iq.harvard.edu/stat110/home