some probability 101 help (will give $)

[thedarknight]

I figure this is a good forum to ask for some math help.

1) Supposed Y~Exp (5). Find the mean, median, and mode of this random variable. (I believe this is a gamma distribution. In order to find the mode, you need to do find the extremum of the density function.)

2) A manager of a store r reports that the time a custome on the second floor must wait for the elevator has a uniform distribution ranging from 1 to 5 minutes. If it takes the elevator 30 seconds to go from floor to floor, find the probability that a hurried customer can reach the first floor in 2.25 minutes after pushing the elevator button on the second floor. (Some integration obviously needs to be done here, I just don't know what to effing integrate. The formula for continuous uniform probability distribution on interval ((y1, y2) is 1/(y2-y1)).

3) A store owner has a daily demand Y for a certain brand of candy sold from bulk bins. Owner will never stock more than one crateful of of this candy and that a crate can fill 50 bins. We measure Y in terms of fractions of a crate. With these units, Y has density function:

f(y)= 3y^2 if 0 < y <1 ( this is greater than or equal to, less than or equal to)
0 otherwise

Grocer can buy a bin's worth of candy for 0.9 dollars and sell bin's worth for $2.25. What amount of candy, C, in bins, should store owner purchase to maximize expected daily profit?

4) Your company is making a sealed bid for a construction project. If your firm has the lowest bid, you will pay another firm (a subcontractor) 170,000 to do the work. If you believe that the minimum bid (in thousands of dollars) of the other firms bidding on the project is a random variable that is uniformly distributed on [140,220] how much should you bid to maximize your expected profit?


I doubt anyone will take a look at this. But this is an online assignment and I will get to see if I have answered correctly. To offer any "motivation" I will give anyone $20 for each question.

Thanks for everyone's help.

[Bantam222]

#3=28?

[ncray]

Question 1: Writing Y~Exp(5) usually means that Y is distributed according to the exponential distribution with parameter lambda = 5. The mean is then 1/5, median ln(2) / 5, and mode 0.
http://en.wikipedia.org/wiki/Exponential_distribution

[crazy canuck]

1 I guess they mean exponential distribution with parameter 5 (gamma has 2 parameters).

Then mean =1/5

mode = 0

median = 0.693/5=0.1386


exponential

[ncray]

Question 2: We need to find the probability that the elevator will arrive within 1.75 (2.25-.5) minutes after the customer pushes the button. Since it follows a uniform distribution, it's just (1.75 - 1)/5 = .15

Edit: The denominator should be 4. The guy below me answered first.

[crazy canuck]

2) Essentially the are asking what is the probability that the elevator will arrive in 1.75 minutes...subtract travel time.

P(X<1.75) = (1.75-1 )/(5-1) = 0.1875

[ncray]

Question 4: Our profit is given by x - 170000, where x is our bid price. We will bid the minimum with probability (220000-x)/80000, so our expected profit is given by (220000-x)/80000 * (x - 170000). We can take the derivative and solve for 0, getting that x = 195000 maximizes our expected profit.

[ncray]

Question 3: Let x = the fraction of a crate that we choose to stock. Then, our profits = $2.25*50 * integral from 0 to x of 3y^2 dy - $.90*50*x, = 112.5x^3 - 45x. We want to maximize this function, where x is from 0 to 1. Looking at the graph, the maximum occurs at x = 1, so we should buy 1 crate = 50 bins of candy.

[thedarknight]

i haven't looked at the answers but i eventually got to figure them out. let's how you guys did.

[thedarknight]

[ QUOTE ]
#3=28?

[/ QUOTE ]

no, you have to establish a revenue and cost function and then integrate from 0 to X and X to 1.