---
# System prepended metadata

title: 'Further Topics on RVs: Derived Distributions'
tags: [Probability]

---

---
tags: Probability
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# Further Topics on RVs: Derived Distributions

## Two-step approach to Calculating Derived PDF

1. Calculate the PDF $F_Y(y)$ of $Y$ using the formula
$$
F_Y(y) = \mathbf{P}(g(X) \leq y) = \int_{\{x|g(x) \leq y\}}f_X(x) \, \mathrm{d}x
$$
2. Differentiate to obtain the PDF (called the derived distribution) of $Y$
$$
f_Y(y) = \frac{\mathrm{d}F_Y(y)}{\mathrm{d}y}
$$

:::spoiler Example 4.1
![](https://i.imgur.com/m8jHRwH.png)
:::

:::spoiler Example 4.3
![](https://i.imgur.com/zxNkGmc.png)
:::

## The PDF of a Linear Function of a RV

* Let $X$ be a continuous rv with PDF $f_X(x)$, and let $Y = aX + b$, for some scalar $a \not= 0$ and b, then
$$
f_Y(y) = \frac{1}{|a|}f_X(\frac{y - b}{a})
$$
  ![](https://i.imgur.com/KFmRnyI.png)
  :::spoiler Proof
  ![](https://i.imgur.com/6JpDC9u.png)
  :::

:::spoiler Example 4.4
![](https://i.imgur.com/43BNQ2X.png)
:::