1. ## Joint Probability

Hi
I've been looking at this problem for awhile now.

Annie and Alvie have agreed to meet between 5:00 pm and 6:00 pm for dinner at a local health food restaurant. Let X=Annie's arrival time and Y=Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6].

a. What is the joint pdf of X and Y?
b. What is the probability that they both arrive between 5:15 and 5:45?
c. If the first one to arrive will wait only 10 minutes before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [Hint: The event of interest is A={(x,y): | x-y | ≤ 1/6}.

For [A]

I set the pdf of x & y = (1/60) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

for the pdf of f(x,y) = (1/3600) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

Is this correct? If not, please explain.

Can you also show me how I can set up parts [B] & [C]

2. Originally Posted by shogunhd
Hi
I've been looking at this problem for awhile now.

Annie and Alvie have agreed to meet between 5:00 pm and 6:00 pm for dinner at a local health food restaurant. Let X=Annie's arrival time and Y=Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6].

a. What is the joint pdf of X and Y?
b. What is the probability that they both arrive between 5:15 and 5:45?
c. If the first one to arrive will wait only 10 minutes before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [Hint: The event of interest is A={(x,y): | x-y | ≤ 1/6}.

For [A]

I set the pdf of x & y = (1/60) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

for the pdf of f(x,y) = (1/3600) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

Is this correct? If not, please explain.

Can you also show me how I can set up parts [b] & [C]
Since you're told that X and Y are:

1. independent, and
2. each is uniformly distributed on the interval [5, 6]

the joint pdf is $f(x, y) = g(x) g(y) = \left( \frac{1}{6 - 5} \right) \left( \frac{1}{6 - 5} \right) = 1$ for $5 \leq x \leq 6$, $5 \leq y \leq 6$ and zero otherwise.

I'm out of time now - will come back later to look at (b) and (c) unless someone else replies in the meantime.

3. Originally Posted by shogunhd
Hi
I've been looking at this problem for awhile now.

Annie and Alvie have agreed to meet between 5:00 pm and 6:00 pm for dinner at a local health food restaurant. Let X=Annie's arrival time and Y=Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6].

a. What is the joint pdf of X and Y?
b. What is the probability that they both arrive between 5:15 and 5:45?
c. If the first one to arrive will wait only 10 minutes before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [Hint: The event of interest is A={(x,y): | x-y | ≤ 1/6}.

For [A]

I set the pdf of x & y = (1/60) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

for the pdf of f(x,y) = (1/3600) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

Is this correct? If not, please explain.

Can you also show me how I can set up parts [b] & [C]
Originally Posted by mr fantastic
Since you're told that X and Y are:

1. independent, and
2. each is uniformly distributed on the interval [5, 6]

the joint pdf is $f(x, y) = g(x) g(y) = \left( \frac{1}{6 - 5} \right) \left( \frac{1}{6 - 5} \right) = 1$ for $5 \leq x \leq 6$, $5 \leq y \leq 6$ and zero otherwise.
[snip]
b. $\Pr(1/4 \leq X \leq 3/4, 1/4 \leq Y \leq 3/4) = \int_{y = 1/4}^{y = 3/4} \int_{x = 1/4}^{x = 3/4} 1 \, dx \, dy = ....$

c. You need to integrate f(x, y) over the region $R_{xy}$ which I'll describe below (I'm too lazy to draw a diagram etc.):

Consider the square in the xy-plane bounded by the coordinate axes and the lines y = 1, x = 1. Note that |x - y| = 1/6 => y = x - 1/6 or y = x + 1/6. The region $R_{xy}$ is the area of the square that is between these two lines ....

If you have trouble setting up the necessary double integral, please say so. To do it you'll need to divide the region into three subregions .....

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### if the first one to arrive will wait only 20 min before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [hint: the event of interest is

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