Probability Distributions: Uniform and Normal Random Variables

Probability Distributions

Uniform Random Variables

The uniform distribution is a type of probability distribution in which all outcomes are equally likely within a specified interval. It is commonly used to model random variables that have constant probability over a range.

• Definition: A random variable X is uniformly distributed on the interval [a, b] if its probability density function (PDF) is constant for all x in [a, b].

• Probability Density Function (PDF):

$f(x) = \frac{1}{b-a}, \quad a \leq x \leq b$

• Mean (Expected Value):

$E[X] = \frac{a+b}{2}$

• Variance:

$Var(X) = \frac{(b-a)^2}{12}$

• Example: If X is uniformly distributed between 20 and 50, then:

$a = 20, \quad b = 50$

$E[X] = \frac{20+50}{2} = 35$

$Var(X) = \frac{(50-20)^2}{12} = \frac{900}{12} = 75$

• Probability Calculation: To find $P(c \leq X \leq d)$ for $a \leq c < d \leq b$:

$P(c \leq X \leq d) = \frac{d-c}{b-a}$

• Application: Uniform distributions are used in simulations, random sampling, and modeling scenarios where each outcome in an interval is equally likely.

Normal (Gaussian) Random Variables

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by its bell-shaped curve. It is widely used in statistics due to the Central Limit Theorem and its prevalence in natural phenomena.

• Definition: A random variable X is normally distributed with mean μ and standard deviation σ if its PDF is:

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

• Standard Normal Distribution: When $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution.

• Z-score: The z-score is used to standardize values:

$z = \frac{x - \mu}{\sigma}$

• Probability Calculation: Probabilities are found using the cumulative distribution function (CDF) or standard normal tables.

• Example: If $X \sim N(30, 5^2)$, then $\mu = 30$, $\sigma = 5$.

• To find $P(X > 35)$:

$z = \frac{35 - 30}{5} = 1$

$P(X > 35) = 1 - P(Z < 1) = 1 - 0.8413 = 0.1587$

• Application: Normal distributions are used in measurement errors, heights, IQ scores, and many natural and social phenomena.

Comparison: Uniform vs. Normal Distribution

Below is a table comparing key properties of the uniform and normal distributions:

Property	Uniform Distribution	Normal Distribution
Shape	Rectangular (constant probability)	Bell-shaped (symmetric)
Parameters	Lower bound (a), Upper bound (b)	Mean (μ), Standard deviation (σ)
Mean	$\frac{a+b}{2}$	$\mu$
Variance	$\frac{(b-a)^2}{12}$	$\sigma^2$
Probability Calculation	Proportional to interval length	Requires integration or standard normal tables
Applications	Random sampling, simulations	Measurement errors, natural phenomena

Standardization and Z-scores

Standardization is the process of converting a normal random variable to a standard normal variable using the z-score formula. This allows for the use of standard normal tables to find probabilities.

• Formula:

$z = \frac{x - \mu}{\sigma}$

• Example: For $X \sim N(30, 5^2)$, to find $P(X < 25)$:

$z = \frac{25 - 30}{5} = -1$

$P(X < 25) = P(Z < -1) = 0.1587$

Summary of Key Formulas

• Uniform Distribution:

  • PDF: $f(x) = \frac{1}{b-a}$

  • Mean: $E[X] = \frac{a+b}{2}$

  • Variance: $Var(X) = \frac{(b-a)^2}{12}$

  • Probability: $P(c \leq X \leq d) = \frac{d-c}{b-a}$

• Normal Distribution:

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

  • Z-score: $z = \frac{x - \mu}{\sigma}$

Additional info:

• Some calculations and notation in the original file were unclear; standard formulas and examples have been provided for completeness.

• Applications and context for both distributions have been expanded for academic clarity.