KA

Lecture 6

6.0 Calculus Review

Continuous Random Variables

Indefinite Integrals:

An indefinite integral represents a family of functions and is denoted mathematically as:

[ \int f(x) , dx + C = F(x) ]

Where:

  • F(x): Represents the antiderivative of f(x).

  • C: A constant representing that there are infinitely many antiderivatives, differing by constant values.

  • Therefore, ( F'(x) = f(x) ) indicates the relationship of differentiation to the integral.

Definite Integrals:

The definite integral of a function f(x) over the interval [a, b] gives the net area between the function and the x-axis and is expressed as:

[ \int_{a}^{b} f(x) , dx = F(b) - F(a) ]

This calculation uses the Fundamental Theorem of Calculus, showing the relationship between differentiation and integration.

6.1 Uniform Probability Distribution

General Characteristics:

A uniform probability distribution describes a situation where all outcomes are equally likely, resulting in a constant probability across a defined range.

Example - Slater’s Buffet:

Let's consider the scenario of a salad quantity taken by customers, which is uniformly distributed between 5 oz and 15 oz.

Probability Calculation:

To find the probability of a customer taking between 12 oz and 15 oz, calculate the area under the probability density function (pdf) for this interval:

[ P(12 < X < 15) = \int_{12}^{15} f(x) , dx = \frac{3}{10} = 0.3 ]

Where:

  • X: Represents the random variable of salad quantity.

  • f(x): The probability density function across the defined range.

Calculation of Expected Value and Variance:

In this uniform distribution scenario:

6.2 Normal Probability Distribution

Overview:

The normal distribution, often referred to as the Gaussian distribution, is a fundamental concept in probability and statistics, frequently encountered in various statistical analyses and natural phenomena.

Defined by Two Parameters:

  • Mean (μ): The average of the distribution, determining its center.

  • Standard Deviation (σ): Measures the spread or dispersion of the distribution around the mean.

Normal Probability Density Function:

The formula for the normal distribution is: [ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} ] Where:

  • e: The base of the natural logarithm (approximately equal to 2.71828).

  • x: The variable for which the probability is being calculated.

Properties:

  • The total area under the curve equals 1, confirming that the sum of probabilities of all outcomes is complete.

  • The mean, median, and mode are all equal in a normal distribution, signifying its symmetry.

Empirical Rule:

  • Approximately 68.26% of data falls within one standard deviation (μ ± σ).

  • About 95.44% falls within two standard deviations (μ ± 2σ).

  • Nearly 99.72% lies within three standard deviations (μ ± 3σ).

Example - Pep Zone:

Assuming demand for a product is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons, we seek the probability of a stockout when the stock is 20 gallons.

Z-score Calculation:

Convert to z-score using the formula: [ z = \frac{x - \mu}{\sigma} = \frac{20 - 15}{6} = 0.83 ]

Probability from Z-table:

Find [ P(z \leq 0.83) = 0.7967 ]

Stockout Probability:

The probability of stockout is therefore: [ 1 - P(z \leq 0.83) = 0.2033 ]

6.4 Exponential Probability Distribution

Characteristics:

The exponential probability distribution is mainly used for modeling the time until an event occurs, such as the waiting time before a Poisson process. The mean and standard deviation are equal in this distribution, creating a significant relationship between these two measures.

Probability Density Function:

The probability density function is expressed as: [ f(x) = \frac{1}{\mu} e^{-\frac{x}{\mu}} ] Where:

  • μ: The mean or expected value of the distribution.

Cumulative Probability:

The cumulative distribution function is given by: [ P(X \leq x) = 1 - e^{-\frac{x}{\mu}} ]

Example - Al’s Full-Service Pump:

Consider Al’s establishment, which experiences an average mean time of 3 minutes between customer arrivals.

Probability Calculation:

To find the probability that a car arrives in 2 minutes or less, use the cumulative function to determine the corresponding threshold for this scenario.

Threshold Time Calculation:

To find the time after which the probability of arrival equals 0.6, solve: [ P(X \leq x) = 0.6 ] resulting in [ x = 2.75 ] minutes.