Special Probability Density Functions: Uniform and Exponential Distributions

Uniform Distribution

  • Definition and Probability Density Function:   f(x)=1b−af(x) = \frac{1}{b - a} for x∈[a,b]x \in [a, b]

  • Expected Value (Mean):   μ=12(b+a)\mu = \frac{1}{2}(b + a)

  • Variance:   Var(X)=112(b−a)2\text{Var}(X) = \frac{1}{12}(b - a)^2

  • Standard Deviation:   σ=112(b−a)\sigma = \frac{1}{\sqrt{12}}(b - a)

  • Example (Daily Insurance Sales Uniformly Distributed on [10,60][10, 60] in Thousands of Dollars):

    • Mean daily sales:     μ=12(60+10)=35\mu = \frac{1}{2}(60 + 10) = 35 (representing $35,000\$35,000)

    • Probability of sales exceeding $50,000\$50,000 (X>50X > 50):     P(X>50)=∫5060150 dt=0.2P(X > 50) = \int_{50}^{60} \frac{1}{50}\,dt = 0.2

Exponential Distribution

  • Definition and Probability Density Function (Used for modeling decreasing likelihood over time):   f(x)=ae−axf(x) = a e^{-a x} for x∈[0,∞)x \in [0, \infty) with constant a>0a > 0

  • Total Probability Verification:   ∫0∞ae−ax dx=lim⁡b→∞(−e−ab+1)=1\int_{0}^{\infty} a e^{-a x}\,dx = \lim_{b \to \infty} \left( -e^{-a b} + 1 \right) = 1

  • Expected Value (Mean):   μ=1a\mu = \frac{1}{a}

  • Variance:   Var(X)=1a2\text{Var}(X) = \frac{1}{a^2}

  • Standard Deviation:   σ=1a\sigma = \frac{1}{a}

  • Example (Consecutive Dry Days in Regions of Spain with Mean μ=8 days\mu = 8\,\text{days}):

    • Rate parameter:     a=18=0.125a = \frac{1}{8} = 0.125

    • Probability density function:     f(t)=18e−0.125tf(t) = \frac{1}{8} e^{-0.125 t} for t≥0t \ge 0

    • Probability of 1010 or more consecutive dry days:     P(X≥10)=∫10∞18e−0.125t dt=0.2865P(X \ge 10) = \int_{10}^{\infty} \frac{1}{8} e^{-0.125 t}\,dt = 0.2865

    • Probability of fewer than 22 dry days:     P(X<2)=∫0218e−0.125t dt=0.2212P(X < 2) = \int_{0}^{2} \frac{1}{8} e^{-0.125 t}\,dt = 0.2212