Q1) Suppose you have a random variable X, where you know the following percentiles:
- p10=8
- p25=73
- p75=98
- p90=111
Further, denote the cumulative density function for X by F. Which of the following is true?
F(98)=0.25
1-F(111)=0.1
F(10)=0.8
F(0.25)=73
Q2)
Suppose you have a random variable A which is the sum of 9 squared IID N(0,1) random variables and another random variable B which is the sum of 25 squared IID N(0,1) random variables. Further, denote Z to be an N(0,1) random variable. Define two additional random variables as follows:
- X=Z/sqrt{A/9}
- Z=(A/9)/(B/25)
where sqrt{} is the square root function. Denote the Normal, Chi-square, t, and F distributions to be N(mu,sigma), ChiSq(df), t(df), and F(df1,df2), respectively. Let "~" further denote "is distributed." What are the distributions of A, X, and Z?
A~F(9,25), X~t(9), Z~F(3,5)
A~ChiSq(3), X~t(5), Z~F(9,25)
A~ChiSq(9), X~N(0,1), Z~F(3,5)
A~ChiSq(9), X~t(9), Z~F(9,25)
