Lecture 5 - Continuous Distribution

Continuous Probability Distributions

  • Focus on continuous random variables, which can take infinite values within an interval.

Probability Density Function

  • An equation used to compute probabilities for continuous random variables.

  • Must satisfy two rules:

    • The total area under the curve equals 1.

    • The height (f(x)) must be ≥ 0 for all values.

  • Probability of a variable X falling in the interval [a, b] is calculated using the integral of the pdf:

    • ( P(a \leq X \leq b) = \int_{a}^{b} f(x) dx )

Continuous Uniform Distribution

  • All values in a given interval are equally likely.

  • Defined by the constant pdf:

    • ( f(x) = \frac{1}{b-a} )

  • Mean (( \mu = \frac{a+b}{2} )), Variance (( \sigma = \frac{(b-a)^2}{12} ))

Normal Probability Distribution

  • Characterized by a symmetric bell-shaped curve.

  • Mean, median, and mode are equal.

  • Properties include:

    • Symmetric about the mean.

    • Single peak at the mean (( \mu )).

    • Inflection points at ( \mu - \sigma ) and ( \mu + \sigma ).

    • Total area is 1, and area above/below the mean is 0.5.

    • Follows the Empirical Rule (68-95-99.7).

Standardizing the Normal Curve

  • The Z-score standardizes a normal variable:

    • ( z = \frac{x - \mu}{\sigma} )

  • Allows finding areas under the curve to determine probabilities.

Empirical Rule for Bell-Shaped Distribution

  • Estimates percentages of data within k standard deviations from the mean:

    • 68% within 1 sigma, 95% within 2 sigma, 99.7% within 3 sigma.

Binomial Probability Distribution

  • Experiments involve:

    1. A fixed number of trials (n).

    2. Independent trials.

    3. Two outcomes: success (p) and failure (1-p).

    4. Same probability of success across trials.

  • For large n, the distribution approaches normality when ( np(1-p) \geq 10 ).

Normal Approximation to the Binomial Distribution

  • If conditions are met, the binomial random variable can be approximated by a normal variable:

    • ( \mu_x = np )

    • ( \sigma_x = \sqrt{np(1-p)} )

Gamma Distribution

  • A continuous distribution modeling time until α successes occur.

  • Probability density function:

    • ( f(x; \lambda) = \begin{cases} \frac{1}{\beta^\alpha \Gamma(\alpha)} x^{\alpha-1} e^{-x/\beta} & x \geq 0 \ 0 & \text{otherwise} \end{cases} )

  • Mean ( \mu = \alpha \beta ) and Variance ( \sigma^2 = \alpha \beta^2 ).

Exponential Distribution

  • Special case of the gamma distribution where ( \alpha = 1 ).

  • Characterizes time until an event occurs with mean ( \mu = \frac{1}{\lambda} ) and variance ( \sigma^2 = \frac{1}{\lambda^2} ).