Need help with statistics?

Question

Help me with the questions. Please try to explain.
The question is at the website and the link is:

http://img2.freeimagehosting.net/image.php?5eb6e4ec57.jpg

http://www.supload.us/free/Scan10013-20070526020457.jpg/view/

Answer 1

p(r+1) = e^(-a)(a^(r+1))/(r+1)!
p(r) = e^(-a)(a^(r))/(r)!
ratio = a/(r+1) ( i used a instead of lambda, since I do not know how to get the symbol.
p(r+1)>p(r) implies the ratio is greater than 1 implies a/(r+1)>1 implies a>r+1 implies r< a-1. QED.
When a = 12.3, r should be smaller than 11.3 for p(r+1) to be larger than p(r) The larget number for which p(r+1) to be greater than p(r) is 11. ie p(r=12) is the largest value. Thus p(1)
Nice question, and you should put the effort to doing your work, thus feeling a greater sense of accomplishment and learn more when done.

What you learn not from personal effort, you will forget.
Edit: I recorrect myself: the probability is maximized when r =12. I apolgize. When I saw the solution of the other guy, i rethought my answer and went to manually try to verify which probability is largest, but i used lambda = 11.3 by mistake instead of 12.3. The correct answer is 12!!!

Answer 2

p(r,λ) = (e^-λ)(λ^r)/r!
p(r + 1)/p(r) = [(e^-λ)(λ^(r + 1)/(r + 1)!]/(e^-λ)(λ^r)/r!
p(r + 1)/p(r) = (λ^(r + 1))r!/((r + 1)!(λ^r))
p(r + 1)/p(r) = (λ^(r + 1))/(λ^r)(r!/(r + 1)!
p(r + 1)/p(r) = λ/(r + 1)

p(r + 1) = λp(r)/(r + 1)
r + 1 = λp(r) / p(r + 1)
r = λp(r) / p(r + 1) - 1
if r < λ - 1
λp(r) / p(r + 1) - 1 < λ - 1
λp(r) < λ p(r + 1)
p(r) < p(r + 1)

ii r = int(12.3 - 1) = 11
iii r = 15 - 1 = 14