Question

Consider a continuous random variable X with the probability density function f X ( x ) = x/C , 3 ≤ x ≤ 7, zero elsewhere. Consider Y = g( X ) = 100/(x^2+1). Use cdf approach to find the cdf of Y, FY(y). Hint: F Y ( y ) = P( Y <y ) = P( g( X ) <y ) =

Answer 1

f_X(x)=x/c ,3 x 7 ,zero elsewhere

first we find value of c ;

  

as 20/c=1 => c=20

Now we find range of Y-

3x7

9x^249 squaring both sides

10x^2+150 adding 1 to each

1/501/(x²+1)1/10 taking reciprocal

2100/(x^2+!)10 multiplying by 100

hence 2g(x)10.or 2Y10

Now find CDF

F_Y(y)=P(Yy)

=P(100/x^2+1 y) = P(100/y x^2+1) [as both y and (x^2+1) is positive ]

= P[(100/y)-1x^2] [take 1 on the other side]

= P[((100-y)/y)^1/2x [here x is positive so we do not have to worry about negative values of x]

= [take 1/2 outside and put limits]

=     

we have already found the range of Y i.e 2Y10.