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 with constant probability over a range.

• Definition: A random variable X is said to be uniformly distributed over the interval [a, b] if its probability density function (pdf) is constant within that interval.

• 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:

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

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

  • Probability that X is less than 30: $P(X < 30) = \frac{30-20}{50-20} = \frac{10}{30} = 0.333$

• Applications: 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, which states that the sum of many independent random variables tends toward a normal distribution.

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

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

• Standard Normal Distribution: When μ = 0 and σ = 1, the distribution is called the standard normal distribution. The random variable is often denoted as Z.

• Properties:

  • Symmetric about the mean

  • Mean, median, and mode are equal

  • Approximately 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three (the "empirical rule")

• Z-Score: The z-score measures how many standard deviations an element is from the mean:

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

• Example: If X is normally distributed with mean 30 and standard deviation 5, the probability that X is less than 35 is:

  • Calculate z-score: $z = \frac{35-30}{5} = 1$

  • Look up $P(Z < 1)$ in standard normal tables: $P(Z < 1) \approx 0.8413$

• Applications: Normal distributions are used in measurement errors, natural phenomena, and inferential statistics.

Comparing Uniform and Normal Distributions

Both uniform and normal distributions are continuous, but they differ in shape and properties.

Property	Uniform Distribution	Normal Distribution
Shape	Rectangular (constant probability)	Bell-shaped (peaked at mean)
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 z-tables

Practice Problems and Applications

• Uniform Distribution:

  • Given X ~ Uniform(20, 50), find $P(X < 30)$, mean, and variance.

  • Given X ~ Uniform(0, 10), find $P(3 < X < 7)$.

• Normal Distribution:

  • Given X ~ N(30, 5), find $P(X < 35)$ and $P(25 < X < 40)$.

  • Convert raw scores to z-scores and use standard normal tables to find probabilities.

Additional info: Some content was inferred and expanded for clarity, including definitions, formulas, and examples. The original file contained fragmented homework questions and calculations related to uniform and normal distributions.