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$X$ is a random variable uniformly distributed on the real interval $[0,1]$.

Through some experimentation, I found that the probability density function, PDF of:

$X$ is $1$ or $\dfrac{d}{dx}X$

$2X$ is $\frac{1}{2}$ or $\dfrac{d}{dx}X/2$

$3X$ is $\frac{1}{3}$ or $\dfrac{d}{dx}X/3$

$X^2$ is $\frac{1}{2\sqrt{X}}$ or $\dfrac{d}{dx}\sqrt{x}$

$X^3$ is $\frac{1}{3x^{2/3}}$ or $\dfrac{d}{dx}\sqrt[3]{x}$

The PDF is useful in answering questions such as what is the mean of $X^3$ or what is the probability that $0<2x<\frac{1}{21}$?

  1. How do I find the PDF of functions in general, something like $X+X^3$?

  2. Also, when there is another variable involved, say Y that is a random variable uniformly distributed on the real interval [0,2], how do I find the PDF of expressions like $X+Y^2$ or $XY^2$? This is again most helpful in finding answers like what is the variance of $X+Y^2$ or what is the probability that $XY^2 > 1$?

  3. What if X and Y are not uniformly distributed, but follows some continuous distribution like the Poisson or Gaussian? How do I find the PDFs in this case?

Falrach
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hollow7
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1 Answers1

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You have just discovered that the cumulutative distribution function of an $f(X)$ when $f$ is an invertible monotonuous increasing function can be computed as:

$$\mathbb{P}(f(X)<y)=\mathbb{P}(X<f^{-1}(y)) \; .$$

Raskolnikov
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    It is conventional to use capital $F$ for the cumulative distribution function and lower-case $f$ for the density function. And I don't think it's a good idea to use the same letter, capital $X$, to refer both to the random variable and to the argument to the CDF. – Michael Hardy Mar 25 '12 at 14:57
  • $F_Y(y)=\mathbb{P}(Y<y)$ – Raskolnikov Mar 25 '12 at 14:59