Unit II Uniform, normal, log normal, Exponential & Gamma Distributions

Lecture Notes on Probability Distributions for Continuous Random Variables

Probability distributions for continuous random variables – Uniform, normal, log normal, exponential and Gamma distributions – Applications

Introduction

Probability distributions for continuous random variables play a fundamental role in probability and statistics. In this lecture, we will explore five essential continuous distributions:

• Uniform Distribution
• Normal Distribution
• Log-normal Distribution
• Exponential Distribution
• Gamma Distribution

Each of these distributions has important applications in civil engineering, including material strength analysis, reliability assessment, and environmental studies.

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1. Uniform Distribution

Definition

A random variable $X$ follows a uniform distribution on the interval $(a, b)$ if its probability density function (PDF) is:

$$f(x) = \begin{cases} \frac{1}{b-a}, & a \leq x \leq b \\ 0, & \text{otherwise} \end{cases}$$

Cumulative Distribution Function (CDF)

$$F(x) = \begin{cases} 0, & x < a \\ \frac{x-a}{b-a}, & a \leq x \leq b \\ 1, & x > b \end{cases}$$

Mean and Variance

$$E[X] = \frac{a+b}{2}, \quad Var(X) = \frac{(b-a)^2}{12}$$

Application in Civil Engineering

• Traffic Flow Analysis: Modeling vehicle arrival times at an intersection.
• Construction Materials: Estimating the uniform spread of loads on a beam.

Example Problem

A pedestrian waits for a bus that arrives uniformly between 10:00 and 10:30 AM. What is the probability they wait less than 10 minutes?

Solution: Given $a = 0$ min and $b = 30$ min, the probability of waiting less than 10 minutes is:

$$P(X < 10) = \frac{10 - 0}{30 - 0} = \frac{10}{30} = \frac{1}{3}$$

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2. Normal Distribution

Definition

A random variable $X$ follows a normal distribution with mean $\mu$ and variance $\sigma^2$, denoted as $X \sim N(\mu, \sigma^2)$, if its PDF is:

$$f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}, \quad -\infty < x < \infty$$

CDF

There is no closed-form expression, but values are found using normal tables or statistical software.

Mean and Variance

$$E[X] = \mu, \quad Var(X) = \sigma^2$$

Application in Civil Engineering

• Material Strength: Concrete strength follows a normal distribution.
• Bridge Load Analysis: The distribution of vehicle weights on a bridge.

Example Problem

The compressive strength of concrete follows $N(30, 4^2)$. What is the probability that a randomly selected sample has strength above 35 MPa?

Solution: Standardizing:

$$Z = \frac{X - \mu}{\sigma} = \frac{35 - 30}{4} = \frac{5}{4} = 1.25$$

From the normal table: $P(Z > 1.25) = 1 - 0.8944 = 0.1056$. Thus, the probability is 10.56%.

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3. Log-normal Distribution

Definition

A random variable $X$ follows a log-normal distribution if $Y = \ln(X)$ follows a normal distribution.

PDF:

$$f(x) = \frac{1}{x \sigma \sqrt{2\pi}} e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}}, \quad x > 0$$

Application in Civil Engineering

• Soil Permeability: The distribution of soil permeability measurements.
• Building Service Life: Modeling the lifespan of materials subject to degradation.

Example Problem

If $Y = \ln(X)$ follows $N(2, 0.25)$, find the median of $X$.

Solution: The median of a log-normal distribution is:

$$Median(X) = e^\mu = e^2 = 7.389$$

Thus, the median is 7.39.

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4. Exponential Distribution

Definition

The exponential distribution models the time until an event occurs. The PDF is:

$$f(x) = \lambda e^{-\lambda x}, \quad x > 0$$

Mean and Variance

$$E[X] = \frac{1}{\lambda}, \quad Var(X) = \frac{1}{\lambda^2}$$

Application in Civil Engineering

• Reliability Analysis: Modeling the time between failures of construction equipment.
• Queueing Theory: Modeling the time between vehicle arrivals at toll booths.

Example Problem

A water pump fails on average once every 10 years. What is the probability it lasts more than 15 years?

Solution: Here, $\lambda = \frac{1}{10}$. The probability is:

$$P(X > 15) = e^{-\lambda x} = e^{-\frac{15}{10}} = e^{-1.5} \approx 0.2231$$

So, the probability is 22.31%.

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5. Gamma Distribution

Definition

The gamma distribution is a generalization of the exponential distribution. It is given by:

$$f(x) = \frac{\lambda^k x^{k-1} e^{-\lambda x}}{\Gamma(k)}, \quad x > 0$$

where $k$ is the shape parameter and $\lambda$ is the rate parameter.

Mean and Variance

$$E[X] = \frac{k}{\lambda}, \quad Var(X) = \frac{k}{\lambda^2}$$

Application in Civil Engineering

• Flood Risk Analysis: Modeling the time between extreme rainfall events.
• Structural Load Analysis: Modeling the cumulative load effects over time.

Example Problem

If failures in a bridge structure follow a gamma distribution with $k = 2$ and $\lambda = 1/5$, find the probability that failure occurs before 8 years.

Solution: The cumulative probability is obtained using the gamma CDF:

$$P(X \leq 8) = \int_0^8 \frac{(1/5)^2 x e^{-x/5}}{\Gamma(2)} dx$$

Using tables or software, $P(X \leq 8) \approx 0.798$, or 79.8%.

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Summary Table

Distribution	PDF	Mean	Variance	Civil Engineering Applications
Uniform	$\frac{1}{b-a}$	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$	Traffic flow, material loads
Normal	$\frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$	$\mu$	$\sigma^2$	Concrete strength, vehicle loads
Log-normal	$\frac{1}{x \sigma \sqrt{2\pi}} e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}}$	$e^{\mu + \sigma^2/2}$	$(e^{\sigma^2} -1)e^{2\mu + \sigma^2}$	Soil permeability, material lifespan
Exponential	$\lambda e^{-\lambda x}$	$1/\lambda$	$1/\lambda^2$	Equipment failure, traffic delays
Gamma	$\frac{\lambda^k x^{k-1} e^{-\lambda x}}{\Gamma(k)}$	$k/\lambda$	$k/\lambda^2$	Flood risk, structural loads

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Conclusion

Understanding these distributions helps civil engineers make informed decisions in infrastructure design, reliability analysis, and risk management.