A random variable B is distributed as P{B< x} = 1 − e^(−ax) (x > 0).Find density function of a) sqrt(B) , b) (1/a) ln{B} .
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Tibor Pejić
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a)
First we'll calculate cumulative distribution function of `Y=sqrt(B)` and that can be done this way:
`F_Y(x)=P(sqrt(B) leq x)=P(B leq x^2) = 1-e^(-ax^2)`
The second equality is valid because we are interested only in `x>0.`
Now to get density function of `Y` you simply derivate distribution function.
`f_Y(x)=(dF_Y)/(dx) (x) = 2ae^(-ax^2)x`
b)
This case is very similar, but here we have `Y=1/a ln(B)` so:
`F_Y(x)=P(1/a ln(B) leq x)=P(B leq e^(ax)) = 1-e^(-ae^(ax))`
`f_Y(x)=(dF_Y)/(dx) (x) = a^2 e^(-ae^(ax)+ax)`
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