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.
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.
