SOLUTION: Suppose that Y has density function f(y) = ky(1-y), 0<=y<=1, f(y) = 0, elsewhere. a, find the value of k that makes f(y) a probability density function b, find P(.4<=Y<

Algebra ->  Probability-and-statistics -> SOLUTION: Suppose that Y has density function f(y) = ky(1-y), 0<=y<=1, f(y) = 0, elsewhere. a, find the value of k that makes f(y) a probability density function b, find P(.4<=Y<      Log On


   

Question 331508: Suppose that Y has density function
f(y) = ky(1-y), 0<=y<=1,
f(y) = 0, elsewhere.
a, find the value of k that makes f(y) a probability density function
b, find P(.4<=Y<=1)
c, find P(.4<=Y<1)
d, find P(Y<=.4|Y<=.8)
e, find P(Y<.4|Y<.8).
Answer by jrfrunner(365) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose that Y has density function
f(y) = ky(1-y), 0<=y<=1,
f(y) = 0, elsewhere.
a, find the value of k that makes f(y) a probability density function
----
K* integal (0 to 1)(y(1-y) dy =+K%2A%28+Y%5E2%2F2-y%5E3%2F3%29 (evaluated for y from 0 to 1)
= K*(1/2-1/3)=1
=k/6=1
K=6
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b, find P(.4<=Y<=1)
----
6%2A%28Y%5E2%2F2-Y%5E3%2F3%29 evalutated for y=0.4 to 1
6%2A%281%2F2-1%2F3-%280.4%5E2%2F2-0.4%5E3%2F3%29%29
=6(1/6-16/200+64/3000)=0.648
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c, find P(.4<=Y<1)
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same as b, continuous functions do not have probability (area) at point
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d, find P(Y<=.4|Y<=.8)
----
P(Y<=.4|Y<=.8)=P(Y<=0.4 and Y<=0.8)/P(Y<0.8) = P(Y<=0.4)/P(Y<0.8)
P(y<0.4) = +6%2A%280.4%5E2%2F2-0.4%5E3%2F3+-0%29+=+0.352
P(y<0.8) = 6%2A%280.8%5E2%2F2-0.8%5E3%2F3+-0%29+=+0.896
P(Y<=.4|Y<=.8)=0.352/0.896=0.393
===================
e, find P(Y<.4|Y<.8).
---
same as d