Calculating Mean and Variance for Graduation Ceremony Guest Tickets

[vikkisut88]

Homework Statement

Suppose that, on average, 70% of graduating students want 2 guest tickets for a graduation ceremony, 20% want 1 guest ticket and the remaining10% don't want any guest tickets.

(a) Let X be the number of tickets required by a randomly chosen student. Find the mean and variance of X.

(b) If 500 guest tickets are available for a ceremony at which 300 students are graduating, what is the probability that there will not be enough tickets to satisfy demand?

Homework Equations

Variance = E(X²) - E²(X)

The Attempt at a Solution

I've got the mean to be ((0.7*2) + (0.2*1) + (0.1*0))/3 = 0.53and the variance to be ((0.7²*2) + (0.2²*1) + (0.1²*0))/3 - 0.53² = 0.7391 but this seems wrong because the variance is then bigger than the mean?!

 

[vikkisut88]

Okay I have since realized that for part a) I think i was doing it wrong so now for the mean I have:

((0.7*2) + (0.2*1) + (0.1*0))/3 = 0.53

But for the Variance I have:

((0.7 - 0.53)²+ (0.2 - 0.53)² + (0.1 - 0.53)²)/3 = 0.1076 which makes far more sense!

Now I'm thinking of using the Central Limit Theorem to solve b)??!

 

[HallsofIvy]

Okay I have since realized that for part a) I think i was doing it wrong so now for the mean I have:

((0.7*2) + (0.2*1) + (0.1*0))/3 = 0.53

This is still incorrect. Suppose every student wanted one ticket. Would the average number be ((0.7*1) + (0.2*1) + (0.1*1))/3 = 1/3? Of course not. The ".7", ".2", and ".1" are already "averages". Don't divide by 3: 0.6*2+ 0.2*1+ 0.1*0= 1.4.

But for the Variance I have:

((0.7 - 0.53)²+ (0.2 - 0.53)² + (0.1 - 0.53)²)/3 = 0.1076 which makes far more sense!

There is no reason in the world why the variance can't be larger than the mean.

Now I'm thinking of using the Central Limit Theorem to solve b)??!

 

[vikkisut88]

Right okay, so now for the variance do I use the same formula but use 1.6 (I presume it was just a typo and it should have been 0.7 *2) instead of 0.53?
Do I then use the Central Limit Theorem for part b?