Continuous Probability Distributions

Continuous Probability Distributions

Overview of Continuous Probability Distributions

• Continuous random variables can assume any value within an interval on the real line or in a collection of intervals.

• Probability cannot be assigned to a specific value of a continuous random variable.

• Instead, we calculate the probability of a random variable falling within a specified interval.

Probability within an Interval

• The probability of a continuous random variable taking a value within an interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2

Uniform Probability Distribution

• A random variable is uniformly distributed when the probability is proportional to the length of the interval.

• The uniform probability density function is mathematically expressed as:

  • $f(x) = \frac{1}{b - a}$ for a < x < b

  • $f(x) = 0$ elsewhere

  • Where:

  • $a$ = smallest value the variable can assume

  • $b$ = largest value the variable can assume

Expected Value and Variance of Uniform Distribution

• The expected value of $x$ is calculated using:

  • $E(x) = \frac{(a + b)}{2}$

• The variance of $x$ is defined as:

  • $Var(x) = \frac{(b - a)^2}{12}$

Example of Uniform Probability Distribution

• Case: Slater’s customers charge based on salad taken, uniformly distributed between 5 ounces and 15 ounces.

  • PDF:
    $f(x) = \frac{1}{10}$ for 5 < x < 15
    $f(x) = 0$ elsewhere.

  • Here, $x$ represents the salad plate filling weight.

• Calculations:

  • Expected Value:
    $E(x) = \frac{(5 + 15)}{2} = 10$

  • Variance:
    $Var(x) = \frac{(15 - 5)^2}{12} = 8.33$

Area as a Measure of Probability

• The area under the probability density function graph corresponds directly to the probability.

• This principle applies universally to all continuous random variables.

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Normal Probability Distribution

• The normal probability distribution is the most significant distribution for describing continuous random variables and is widely utilized in statistical inference.

• Applications include:

  • Heights of individuals

  • Rainfall measurements

  • Standardized testing scores

  • Scientific measurement outcomes.

Normal Probability Density Function

• The characteristics include:

  • $μ$ = mean

  • $σ$ = standard deviation

  • The constants are:

  • $\pi = 3.14159$

  • $e = 2.71828$

• The expression for Normal Probability Density Function

  • $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}$

Characteristics of Normal Distribution

• Symmetry: It is symmetric around the mean, which indicates that the skewness measure equals zero.

• Mean, Median, Mode: The highest point on the curve corresponds to the mean (also median and mode).

• Mean Range: The mean can take any numerical value (negative, zero, or positive).

• Standard Deviation Impact: The standard deviation controls the curve's width; larger standard deviations yield wider, flatter distributions.

Probabilities and Distribution Area

• Areas under the normal curve represent probabilities:

  • The entire area under the curve equals 1.

  • The area to the left of the mean is 0.5, and to the right is also 0.5.

Empirical Rule

• For a normal random variable:

  • $68.26\%$ of values fall within +/- 1 standard deviation from the mean.

  • $95.44\%$ of values are within +/- 2 standard deviations from the mean.

  • $99.72\%$ of values are within +/- 3 standard deviations from the mean.

Standard Normal Probability Distribution

• A random variable transformed into a standard normal distribution has a mean of 0 and a standard deviation of 1.

• The variable $z$ denotes the standard normal random variable.

• Conversion formula to the standard normal distribution is:

  • $z = \frac{x - \mu}{\sigma}$

Example of Standard Normal Probability Distribution

• Case: Pep Zone sells automotive supplies, and a stockout occurs when the oil stock drops to 20 gallons.

• Demand during lead-time is normally distributed with:

  • Mean = 15 gallons

  • Standard Deviation = 6 gallons.

• Objective: Calculate the probability of stockout during replenishment, i.e., determine P(x > 20).

Converting to Standard Normal Distribution

1. Calculate $z$ value:

   • $z = \frac{(20 - 15)}{6} = 0.83$

2. Use cumulative probability tables to find:

   • P(z < 0.83) = 0.7967

3. Calculate the stockout probability:

   • P(z > 0.83) = 1 - P(z < 0.83) = 1 - 0.7967 = 0.2033

Determining Reorder Point

• Target probability of stockout during replenishment is to be no more than 0.05.

• Step 1: Find the $z$ value that corresponds to an area of 0.05 in the right tail (complement of 0.95):

• Step 2: Convert $z$ to value of $x$:

  • For the area of 0.05, use the standard normal table to determine the corresponding z:

  • $x = μ + z_{0.05}σ = 15 + 1.645(6) = 24.87$

  • A reorder point of 25 gallons is needed to maintain an acceptable stockout probability.

• Increasing the reorder point from 20 to 25 gallons decreases the stockout probability from approximately 0.20 to 0.05.

Summary of Impact

• Adjusting the reorder point effectively manages stock levels and customer satisfaction by significantly reducing potential stockouts.