ch04 Continuous Random Variables & Prob Distributions-6

Chapter 4 Continuous Random Variables & Probability Distributions

4.1 Probability Distributions and Probability Density Functions

  • Continuous Random Variables: Variables that can take on an infinite number of values within a given range.

  • Probability Density Function (PDF): A function that describes the likelihood of a continuous random variable taking on a specific value. The area under the curve within a certain interval represents the probability that the variable falls within that interval.

4.2 Cumulative Distribution Functions (CDF)

  • Cumulative Distribution Function: A function that describes the probability that a random variable is less than or equal to a certain value.

  • Calculated as the integral of the PDF.

4.3 Mean and Variance of a Continuous Random Variable

  • Mean (Expected Value): Represents the average value of the random variable; for a continuous variable given by the integral of x multiplied by the PDF over its range.

  • Variance: Represents the spread or dispersion of the random variable; calculated as the integral of (x - mean)² multiplied by the PDF over its range.

4.4 Continuous Uniform Distribution

  • Definition: A simple distribution where all outcomes are equally likely within a defined range [a, b]. The shape of the distribution is rectangular.

Example: Waiting Time at a Bus Stop

  • Scenario: Bus arrives every 30 minutes (from 0 to 30 minutes).

  • Typical Wait Time: Mean waiting time calculated as ( \frac{0 + 30}{2} = 15 ) minutes.

  • Standard Deviation Calculation:

    • ( \sigma = \frac{30 - 0}{\sqrt{12}} \approx 8.66 )

  • Probabilities:

    • Wait more than 25 minutes ( P(25 < wait < 30) ): Probability is 0.1667 (using area under PDF).

    • Wait between 10 and 20 minutes ( P(10 < wait < 20) ): Probability is 0.3333.

4.5 Normal Distribution

  • Characteristics:

    • Bell-shaped curve; symmetric around the mean.

    • Mean, median, and mode are equal.

    • Total area under the curve is 1.

    • Asymptotic: Tails approach the axis but never touch.

    • EXCEL: For not given Z GIVES ARES FOR Z

      • = NORM.DIST(X,MEAN, STANDARD DEVIATION, T/F)

        • all examples use true

        • may have to use empirical rule

        • subtract if in between

    • Empirical Rule:

      • 68% of data within one standard deviation from the mean.

      • 95% within two standard deviations.

      • 99.7% within three standard deviations.

        • different normals^

Standard Normal Distribution

  • Special case with mean 0 and standard deviation 1.

  • Z-value formula:

  • Excel: =NORM.S.DIST(Z,TRUE/FALSE) FOR if given Z

    • z<= TRUE

    • Z> TRUE 1- NORM.S.DIST(Z,TRUE)

Finding Probabilities

  • Use of cumulative standard normal distribution (NORM.S.DIST) to calculate probabilities.

  • Example: For Z = 1.5, find ( P(Z \leq 1.5) ).

4.6 Exponential Distribution

  • Definition: Models the time until an event occurs in a Poisson process.

  • EXCEL:=EXPON.DIST(X,P(X),T/F)

    • true was used in all examples

  • Formula:

  • Mean and Variance: Both depend upon the rate parameter ( \lambda )

Applications of Exponential Distribution

  • Useful in modeling waiting times: e.g., time between customer arrivals, or failures of components.

Important Terms and Concepts

  • Continuous Uniform Distribution: All outcomes equally likely across an interval.

  • Normal Distribution: Symmetrical probability distribution; defined by mean and standard deviation.

  • Exponential Distribution: Describes time between events in a Poisson process.