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≤x)=∫−∞xf(v) dvF(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:
      ∫−∞∞f(v) dv=1\int_{-\infty}^{\infty} f(v) \, dv = 1
    3. The derivative of the CDF gives the PDF:
      f(x)=dFdxf(x) = \frac{dF}{dx}
    4. The probability of an interval $[a,b]$ can be computed:
      P(a≤X≤b)=F(b)−F(a)P(a \leq X \leq b) = F(b) - F(a)
    5. For any specific point, the probability is zero:
      P(X=b)=0P(X = b) = 0

Percentiles and Median

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

Mean and Variance of a Continuous Random Variable

  • Mean $\muX$: μ</em>X=E[X]=∫−∞∞xf(x) dx\mu</em>X = E[X] = \int_{-\infty}^{\infty} x f(x) \, dx
  • Variance $\sigma^2X$: σ2</em>X=V[X]=E[X2]−(E[X])2=∫<em>−∞∞(x−μ</em>X)2f(x) dx\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)={1b−a,amp;a≤x≤b 0,amp;otherwisef(x) = \begin{cases} \frac{1}{b-a}, &amp; a \leq x \leq b \ 0, &amp; \text{otherwise} \end{cases}
  • Normal Distribution: Bell-shaped PDF given by:
    f(x)=12πσ2e−(x−μ)22σ2f(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)=1xσ2πe−(ln⁡x−μ)22σ2f(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)=λe−λx,  x≥0f(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)=1Γ(α)βαxα−1e−xβ,  x≥0f(x) = \frac{1}{\Gamma(\alpha) \beta^{\alpha}} x^{\alpha - 1} e^{-\frac{x}{\beta}}, \; x \geq 0
  • Weibull Distribution: PDF:
    f(x)=αβ(xβ)α−1e−(xβ)α,  x≥0f(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))⋅∣h′(h−1(y))∣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.