Engineering Statistics - Continuous Random Variables

Continuous Random Variables

Definition of Continuous Random Variable

• A random variable $X$ is called continuous if it can take any real value in a given range, making the possible values not finite or countable.
• Examples include:
  • Time spent in traffic
  • Body temperature of a patient
  • Velocity of an atom

Probability Density Function (PDF) and Cumulative Distribution Function (CDF)

• The PDF $f(x)$ defines the probability density of the continuous variable.
• The CDF $F(x)$ can be expressed using the PDF:
  $F(x) = P(X \leq x) = \int_{-\infty}^x f(v) \, dv$
• Key properties of the PDF and CDF:
  1. $F(x)$ is a non-decreasing function.
  2. The normalization condition must be satisfied:
     $\int_{-\infty}^{\infty} f(v) \, dv = 1$
  3. The derivative of the CDF gives the PDF:
     $f(x) = \frac{dF}{dx}$
  4. The probability of an interval $[a,b]$ can be computed:
     $P(a \leq X \leq b) = F(b) - F(a)$
  5. For any specific point, the probability is zero:
     $P(X = b) = 0$

Percentiles and Median

• The $p$th percentile, denoted $\eta<em>p$, is the value of $X$ such that: $P(X \leq \eta</em>p) = p$
• The median (50th percentile) is given by:
  $\mu<em>X = \eta</em>{0.5}$

Mean and Variance of a Continuous Random Variable

• Mean $\mu<em>X$: $\mu</em>X = E[X] = \int_{-\infty}^{\infty} x f(x) \, dx$
• Variance $\sigma^2<em>X$: $\sigma^2</em>X = V[X] = E[X^2] - (E[X])^2 = \int<em>{-\infty}^{\infty} (x - \mu</em>X)^2 f(x) \, dx$

Common Continuous Distributions

• Uniform Distribution: PDF is constant over the range $[a,b]$:
  $f(x) = \begin{cases} \frac{1}{b-a}, & a \leq x \leq b \ 0, & \text{otherwise} \end{cases}$
• Normal Distribution: Bell-shaped PDF given by:
  $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$
  where $\mu$ is the mean and $\sigma$ is the standard deviation.
• Lognormal Distribution: PDF:
  $f(x) = \frac{1}{x \sigma \sqrt{2\pi}} e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}}$
  only for $x > 0$.
• Exponential Distribution: PDF:
  $f(x) = \lambda e^{-\lambda x}, \; x \geq 0$
  where $\lambda$ is the rate parameter.
• Gamma Distribution: PDF for shape $\alpha$ and scale $\beta$:
  $f(x) = \frac{1}{\Gamma(\alpha) \beta^{\alpha}} x^{\alpha - 1} e^{-\frac{x}{\beta}}, \; x \geq 0$
• Weibull Distribution: PDF:
  $f(x) = \frac{\alpha}{\beta} \left(\frac{x}{\beta}\right)^{\alpha - 1} e^{-\left(\frac{x}{\beta}\right)^{\alpha}} , \; x \geq 0$

Function of a Random Variable

• If $Y = h(X)$, where $h$ is a differentiable function:
  • The PDF can be derived from the transformation:
    $f<em>Y(y) = f</em>X(h^{-1}(y)) \cdot |h'(h^{-1}(y))|$
• The mean and variance of $Y$ can be derived from the functions of $X$:
  • $E[Y] = E[h(X)]$ and for variance use similar transformations.

Probability Plots

• Useful for checking if a sample comes from a particular distribution by plotting the empirical versus theoretical percentiles.
• If the points form a straight line, the assumption holds true; substantial deviations indicate a poor fit.