If $X$ is a random variable with PDF $f_X(x), x \in \mathbb{R}$, and $Y = e^X$. What is the PDF $f_Y(y), y > 0$?
How can I compute the PDF of $Y$ knowing that of $X$ and knowing how $Y$ depends on $X$?
Probability density function of dependent random variable
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probability
probability-theory
probability-distributions
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density-function
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0Such computations are usually easier to do with the *cumulative distribution function*, from which the probability density function can be obtained by differentiating. – 2017-01-17
1 Answers
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Let $$F_X(x) = \int_{-\infty}^x f_X(t)dt = P(X\le x).$$ Then for $y>0$ we have $$F_Y(y) = P(Y\leq y) = P(e^X\le y) = P(X\le \ln(y)) = F_X(\ln(y)).$$
Then recall that if $$F_Y(y) = P(Y\le y) = \int_{0}^y f_Y(t)dt,$$ we have $f_Y(y)=F_Y'(y) ,$ so $$f_Y(y) = \frac{d}{dy} F_Y(y)=\frac{d}{dy} F_X(\ln(y)) = \frac{1}{y}F_X'(\ln(y)) = \frac{1}{y}f_x(\ln(y)).$$