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Notes › EENG 3421: Probability and Statistics for Engineers Lecture 16

Probability Density Function

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Definition 16.1 (Probability Density Function).

The probability density function of a continuous random variable $X$ is

$$ f_X(x)=\frac{dF_X(x)}{dx} $$
Theorem 16.2 (Basic Properties of a PDF).

For a continuous random variable $X$ with PDF $f_X(x)$

  • nonnegativity
$$ f_X(x)\ge 0 \quad \text{for all } x $$
  • CDF as an integral of the PDF
$$ F_X(x)=\int_{-\infty}^{x} f_X(u)\,du $$
  • total area
$$ \int_{-\infty}^{\infty} f_X(x)\,dx=1 $$

Proof. Since $F_X(x)$ is nondecreasing its derivative is nonnegative The integral formula follows from the definition of derivative and the fundamental theorem of calculus The total area follows by taking $x\to\infty$ in the integral formula and using $F_X(\infty)=1$

Theorem 16.3 (Probability of an Interval Using the PDF).

For $x_1 $$ P[x_1

Proof. Use

$$ P[x_1and substitute

$$ F_X(x)=\int_{-\infty}^{x} f_X(u)\,du $$

Interpretation of PDF

  • Density is not probability at a point
  • Density converts small interval length into probability
Proposition 16.4 (Small Interval Approximation).

Let $x_2=x_1+\Delta$ with $\Delta>0$ small

$$ P[x_1In the limit $\Delta\to 0$

$$ P[x

References

  • Course lecture slides 18–20

Sources

  • Course lecture slides 18–20

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