1. Let Y ~ Binomial(n,p) with pmf pY(y).Obtain the value of y for which pY(y) is maximum.
2. Let Y ~ geometric(p) with pmf pY(y).
(a) Show that the value p = 1/y maximizes the pmf.
(b) Obtain E(1/Y ).
3. Let X be a random variable, and Y = aX + b for non-random values a and b. Using the fact that MY(t) = (e^bt)MX(at) show that V ar(Y ) = (a^2)V ar(X).
Man...I'm so stressed. The textbook is useless and the prof won't answer questions till all 10 questions are handed in tomorrow. These are the last 3 questions I have.
Thanks in advance!
variable which is not something commonly covered in courses.
In this case we know that the pmf is approximatly symmetric and bell shaped
so we would expect its maximum to be near the mean, and with a bit of handwaving
that it is what I gave can be made plausible.
It makes me think that there may be a mistake in the way the problem has
(the next one is even worse, I can't quite figure out what it wants
maximised, it can't be to find the k that maximises the pmf for a fixed p
it must be to find the p that maximises the pmf for fixed k, which at least
is a problem that is susceptable to the use of calculus)
so for any , we have:
The maximum of for fixed occurs when this is equal to zero, or with some rearrangement and cancelling factors of is equivalent to:
which has solution .
(There is a wrinkle that we should worry about briefly, and that is that the maximum could in principle occur when , but in this case these are minima for all , so we need not worry about such posibbilities)
for discrete case
if k is the mode
P(Y=k) >= P(Y= k+1)
and P(Y=k) >= P(Y=k-1)
substitute into the pmf and solve the inequalites for k
some steps are available at mode of the binomial distribution
some information on derivation of mean, variance ,mode , mgf of binomial distribution is available at derivation of mean, variance ,mode , mgf of binomial distribution
a suggestion for question(3) is to expand
MX(t) = 1 +E(X) t/1! +E(X^2) (t^2) / 2! + ....
replace t with (at) to get MX(at)
then use product rule to differentiate MY(t) w.r.t. 't' and put t =0
hopefully you should get
E(Y) = a E(X) + b
diff. MY(t) w.r.t. 't' once more and put t =0 to get E(Y^2)
Thank you very much Captain.
I did question 1 this way, not sure if it's correct or not:
P(Y=y) = pmf of binomial distribution (sorry dunno how to use the math function)
then i take log P(Y=y)
Next I differentiate log P(Y=y), then set it to 0.
Final answer, P=y/n.
Q3: I got confused at how should I differentiate (e^bt)Mx(at), the Mx(at) part.
I've figured out already and was able to prove that Var(Y)=a^2Var(X).
Thank you so much for your help again!
Till next week....the horror starts again!