Chapter 24: Continuous RVs and PDFs

What is a cumulative distribution function?

Cumulative distribution function

The cumulative distribution function (cdf) of a continuous random variable \(X\), is the function \(F_X(x)\), such that for all real values of \(x\), \[F_X(x)= \mathbb{P}(X \leq x) = \int_{-\infty}^x f_X(s)ds\]

Remarks: In general, \(F_X(x)\) is increasing and

• \(\lim_{x\rightarrow -\infty} F_X(x)= 0\)

• \(\lim_{x\rightarrow \infty} F_X(x)= 1\)

• \(P(X > a) = 1 - P(X \leq a) = 1 - F_X(a)\)

• \(P(a \leq X \leq b) = F_X(b) - F_X(a)\)

Let’s demonstrate the CDF with an example

Example 2

Let \(f_X(x)= 2\), for \(2.5 \leq x \leq 3\). Find \(F_X(x)\).

Derivatives of the CDF

Theorem 1

If \(X\) is a continuous random variable with pdf \(f_X(x)\) and cdf \(F_X(x)\), then for all real values of \(x\) at which \(F'_X(x)\) exists, \[\frac{d}{dx} F_X(x)= F'_X(x) = f_X(x)\]

Finding the PDF from a CDF

Example 3

Let \(X\) be a RV with cdf \[F_X(x)= \left\{ \begin{array}{ll} 0 & \quad x < 2.5 \\ 2x-5 & \quad 2.5 \leq x \leq 3 \\ 1 & \quad x > 3 \end{array} \right.\] Find the pdf \(f_X(x)\).

Let’s go through another example (1/7)

Example 4

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

1. Show \(f_X(x)\) is a pdf.

2. Find \(\mathbb{P}(1 \leq X \leq 3)\).

3. Find \(F_X(x)\).

4. Given \(F_X(x)\), find \(f_X(x)\).

5. Find \(\mathbb{P}(X \geq 1 | X \leq 3)\).

6. Find the median of the distribution of \(X\).

Let’s go through another example (2/7)

Example 4.1

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

1. Show \(f_X(x)\) is a pdf.

Let’s go through another example (3/7)

Do this problem at home for extra practice.

Example 4.2

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

2. Find \(\mathbb{P}(1 \leq X \leq 3)\).

Let’s go through another example (4/7)

Example 4.3

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

3. Find \(F_X(x)\).

Let’s go through another example (5/7)

Do this problem at home for extra practice.

Example 4.4

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

4. Given \(F_X(x)\), find \(f_X(x)\).

Let’s go through another example (6/7)

Example 4.5

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

5. Find \(\mathbb{P}(X \geq 1 | X \leq 3)\).

Let’s go through another example (7/7)

Example 4.6

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

6. Find the median of the distribution of \(X\).