MHB What are the Calculations for Exponential Distribution in Bank Arrival Times?

AI Thread Summary
The discussion focuses on calculating various statistics for an exponential distribution representing bank arrival times, specifically with a rate parameter $\lambda = 2$. The expected time between arrivals is correctly identified as 30 minutes. However, the standard deviation calculation is incorrect; the correct standard deviation is 0.5, not 0.25. The probability of arrival within 4 minutes is approximately 0.9996, and the probability for the interval between 2 and 5 minutes is calculated as approximately 0.018270. Overall, the calculations for expected time and probabilities are accurate, while the standard deviation needs correction.
shamieh
Messages
538
Reaction score
0
Let $X$ = the time between two successive arrivals at the drive-up window of a local bank. $X$ has an exponential distribution with $\lambda = 2$. That is the probability density of $X$ is $f(X | \lambda) = \lambda e^{-\lambda x}, X > 0 $ with $\lambda = 2$. Compute the following:

a) The expected time between two successive arrivals.

b) The standard deviation of the time between successive arrivals.

c) $P(X\le4)$

d) $(P(2\le X<5)$

I just need someone to check my work to make sure I'm doing these right.

I think I've got the first part.. would it be

a) $\mu = 1/2 => 30$ minutes or half an hour?

And for b) I got:

b) $\sigma^2 = 1/\lambda^2 = (1/2)^2 = (1/4)^2 = 1/16$
so $\sigma^2 = \sqrt{1/16} => \sigma = .25$ ?

c) $P(X \le 4) = 1 - e^{-2*4} \approx 0.9996$

d) $\int^5_2 2e^{-2x} dx \approx 0.018270$
 
Last edited:
Mathematics news on Phys.org
Hi shamieh,

your answers are correct except for b). For an exponential distributed variable $X$ holds indeed $\mbox{Var}(X) = \frac{1}{\lambda^2} = \frac{1}{4}$ and hence for the standard deviation $\sigma^2 = \sqrt{\frac{1}{4}} = \frac{1}{2}$.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...

Similar threads

Replies
1
Views
2K
Replies
13
Views
2K
Replies
2
Views
1K
Replies
4
Views
2K
Replies
5
Views
2K
Replies
8
Views
1K
Replies
3
Views
4K
Back
Top