Suppose that Y has the density function:

f(y) = ky(1-y) when y is between (or including) 0 and 1 , and is = 0 elsewhere

a) Find the value of k that makes f(y) a probability density function
b) Find P(0.4<=Y<=1)
c) Find P(Y<=0.4|Y<=0.8)
d) Find P(Y<0.4|Y<0.8)

Thanks in advance!

Hello Roadapples,

I will provide "how" to get the answers in addition to the actual
answers. If you want some free software that can solve these kinds of
questions, you may want to check out Maxima - see the references
below.

For f(y) = k * y * (1-y) [range 0,1], 0 otherwise.

(a) A probability function is the integral of range +/- infinity of
f(x) dx = 1 and f(x) >= 0 over the whole range. The integral of f(y)
is 1/6 [according to Maxima], so K must be 6.

(b) P(0.4 <= Y <= 1) is the integral of the function over that range.
Asking Maxima for the answer yields P=81/125 = 0.648.

(c) P(Y<=0.4|Y<0.8) is the probability of P(Y<=0.4) given P(Y<=0.8) is
true. Basically, this is the ratio of the two probabilities. P(Y<=0.4)
= 22/375 = 0.0586...., P(Y<=0.8) = 56/375 = 0.1493.... Or a result of
11/28 = 0.3928571.

(d) P(Y<0.4 | Y<0.8) - perhaps a typo; otherwise the answer for (c) is
"close". Perhaps you are looking for P(A or B), P(A and B), or P(A|B)
with the signs changed. Of course, I could just misunderstand the
notation - please clarify.

Please use a clarification request on (d) and if I've made a mistake
in notation on one of the other questions.

References
Probability Distribution Functions
  http://csep1.phy.ornl.gov/mc/node13.html
Maxima
  http://www.ma.utexas.edu/users/wfs/maxima.html
or
  http://maxima.sourceforge.net/

  --Maniac

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Request for Answer Clarification by roadapples-ga on 10 Feb 2003 18:35 PST

In (d) its supposed to be "less than" without the '=' whereas in (c)
they should both have less than or equal to...

Everything looks great! Thanks!

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Clarification of Answer by maniac-ga on 10 Feb 2003 19:56 PST

Hello Roadapples,

That is what I read. Perhaps the point is to repeat the answer in [c].
For example, if I evaluate
  P(Y<=0.399 | Y<=0.799) = 0.39167
  P(Y<=0.3999 | Y<=0.7999) = 0.3927385
and so on. As you get closer to .4 and .8, the answer given in (C) is
the limit.

  --Maniac

Thanks once again!