pt 6: Exponential Distribution Review

Skewness in Distributions

  • In skewed distributions, the mean is larger than the median.

  • For symmetrical distributions, the mean and median are similar.

  • The range of values for a distribution typically extends from negative infinity to positive infinity, or between the smallest and largest observed values.

Exponential Distribution Review

  • The probability of xx being less than tt is given by: P(x < t) = 1 - e^{-\lambda t}

  • This formula represents the area to the left of tt on a graph of the exponential distribution.

  • The complement rule can be used to find the probability of xx being greater than tt: P(x > t) = e^{-\lambda t}

  • \lambda (lambda) is a rate parameter, and its units must align with the time unit (e.g., weeks).

Percentiles and Exponential Distribution

  • To find the nnth percentile (e.g., 80th percentile), we are looking for the value where the area to the left is equal to n/100n/100.

  • Formula for the nnth percentile (alpha) in an exponential distribution: −1λln(1−α)\frac{-1}{\lambda} ln(1 - \alpha)

    • \alpha represents the percentile value expressed as a decimal (e.g., 80th percentile is 0.8).

  • Example: Finding the 80th percentile with λ=2\lambda = 2:
    −12ln(1−0.8)=−12ln(0.2)≈0.8047\frac{-1}{2} ln(1 - 0.8) = \frac{-1}{2} ln(0.2) ≈ 0.8047

Quartiles and Percentiles

  • Quartiles divide the data into four equal parts.

    • Q1 (25th percentile)

    • Q2 (50th percentile, median)

    • Q3 (75th percentile)

  • Formula for calculating quartiles, Q<em>iQ<em>i: Q</em>i=−1λln(1−α<em>i)Q</em>i = \frac{-1}{\lambda} ln(1 - \alpha<em>i), where α</em>i\alpha</em>i corresponds to the respective quartile's percentile value as a decimal.

  • Example calculations with λ=2\lambda = 2:

    • Q1 (25th percentile): −12ln(1−0.25)=−12ln(0.75)≈0.1438\frac{-1}{2} ln(1 - 0.25) = \frac{-1}{2} ln(0.75) ≈ 0.1438

    • Q2 (50th percentile): −12ln(1−0.5)=−12ln(0.5)≈0.3466\frac{-1}{2} ln(1 - 0.5) = \frac{-1}{2} ln(0.5) ≈ 0.3466

    • Q3 (75th percentile): −12ln(1−0.75)=−12ln(0.25)≈0.693\frac{-1}{2} ln(1 - 0.75) = \frac{-1}{2} ln(0.25) ≈ 0.693

  • Interpretation: 25% of the data lies to the left of Q1, 50% lies to the left of Q2, and 75% lies to the left of Q3.

Expected Value and Variance of Exponential Distribution

  • Expected value (mean): E[X]=1λE[X] = \frac{1}{\lambda}

  • Variance: Var(X)=1λ2Var(X) = \frac{1}{\lambda^2}

  • Standard deviation: σ=1λ\sigma = \frac{1}{\lambda}

  • For the exponential distribution, the expected value and standard deviation are equal.

Example Calculation

  • Given λ=2\lambda = 2:

  • Expected value: E[X]=12=0.5E[X] = \frac{1}{2} = 0.5

  • Variance: Var(X)=122=14=0.25Var(X) = \frac{1}{2^2} = \frac{1}{4} = 0.25

  • Standard deviation: σ=12=0.5\sigma = \frac{1}{2}=0.5