Monday, March 14, 2005

ERBA= Exams Really Bite Ass

Just got out of the ERBA exam. BLECH. It was comprised of three questions, decreasing in difficulty/impossibility from 1 - 3. After doing the post-exam, confidence-shattering exchange of answers, I realized that most everybody had difficulty with the exam. Of course, I was seated next to Yoav, who, upon reading the first question said, "Yeah!" under his breath. (Yoav, if you're reading, know that I am scratching my nose with a specific finger right now.)

This is the first question: "The daily concentration of a certain pollutant in a stream has an exponential distribution with mean value 2 mg/10^3 liter. Suppose that the unacceptable pollution level is 6 mg/10^3 liter. What is the probability that pollution will be unacceptable in a single day?" Okay, if your name isn't Yoav, P-Wheel, Spriros, Matt, or John Wang and you know how to do this, let me know and I'll buy you a beer.

4 comments:

Yoav said...

Mmm, beer. ;)

You want the probability of getting more than 6mg/10^3L or more than 0.006mg/L. That's 1 - probability of getting less than 0.006mg/L.

The probability of getting less than 0.006mg/L is directly from the cumulative density function, which is 1-e^(-lambda*x) for the exponential distribution. Let x = 0.006, lambda = 1 / mean = 1/0.002 = 500, calculate 1-e^(-500*0.006) = 1-e^(-3) = 0.95. That's the probability of getting less than 0.006, so you want 1-0.95 = 0.05, so 5% or so was my answer to 1a.

Yoav said...

1b follows pretty easily from 1a. I use Poisson but I guess binomial can be used as well. Let N=3, find the proability for k=0 and for k=1, and their sum is the probability of at most once during three days. It ends up being virtually 1.

Yoav said...

1c was easy and doesn't depend on 1a or 1b. The mean is given (0.002), and for exponential standard deviation = mean = 0.002. Then Z = (0.006-0.002)/0.002 = 2, and it's a straight lookup in the normal distribution table from there, giving 0.0228 or 2.28% result, significantly smaller than 1a.

Yoav said...

Do I get to pick a type of beer even if I'm wrong? ;)