Module 3: Random Variables and Normal Distributions

A variable is a measurement or characteristic on which individual observations are made. A random variable is a variable whose possible values are outcomes of a random phenomenon. A domain is a set of all possible values a variable can take.

Discrete random variables is a finite set or countably infinite sequence.

p_x(x_i) = P(X = x_I) is called the Probability Mass Function (PMF).

• 0 <= p_x(x_i) <= 1 as it is a probability,
• Sum of PMF for all values of X = 1.

Continuous random variable can lie on a numerical scale, such as all real numbers between (0, +infinity). If we mapped the data to a histogram, we would see the curve begin to smooth as the number of data points approaches infinity.

This density curve, f_x(x), is called the Probability Density Function (PDF).

• f_x(x) >= 0 but can be greater than 1
• The integral of f_x(x) over the domain of X is 1

The Cumulative Distribution Function (CDF) is defined as F_x(X) = P(X <= x)

• Non-decreasing
• The limit toward -infinity is 0, toward +infinity is 1
• For discrete random variables:

• For continuous random variables:

The Expected Value of a random variable is an average of the possible values weighted by their probabilities. Also called mean and denoted by μ.

• For discrete random variables

• For continuous random variables
  

A generalization for the expected values is the r^th moment of a random variables, R(X^r).

• For discrete random variables:

• For continuous random variables:

The first moment of a random variable is the expect value (mean). The r^th moment of a random variable about the mean, also called the r^th central moment, is defined as: E[(X - μ)^r]

• The first central moment = 0
• The second central moment is the variance denoted as 𝜎²

The variance measures the spread around the mean of a random variable: Var(X) = E[(X - μ)²]

• Also equivelent to Var(X) = E(X²) - [E(X)]²
• The standard deviation is the square root of the variance

The normal distribution:

• is a continuous distribution
• can be expressed by a formula
• also called Gaussian distribution
• is a theoretical model for a population distribution that approximates the distribution of a number of measurement variables
• is appropriate for a number of measures, but not all. Only appropriate for some continuous measurements.
• is symmetric about the mean (i.e P(X > μ) = P(X < μ) = .5)
• is completely characterized mean and variance

The 68/95/99 rule:

• 68.25% of the data falls within 1 SD
• 95.45% of the data falls within 2 SD
• 99.74% of the data falls within 3SD

The standard normal random variable, referred to as Z, is in the scale of SD units from the mean.

Z has a μ=0 and SD = 1 we can standardize any normal distribution with:

By converting to Z-scores we can easily compare the probability events in two different normal distribution.

The k^th percentile is defined as the score that holds the k percent of the scores below it. Ex. 90th percentile is the score that has 90% of the scores below it. We can compute percentiles with: X = μ + Z𝜎

The probability density function for normal curves:

And in normal curves when mean is 0 and SD is 1 this can be simplified.

Relevant R Functions

qnorm(percentile, μ, 𝜎) computes percentiles for normal variables

dnorm(x, μ, 𝜎) will return the height of normal density function with a certain mean and SD at point x

pnorm(z, μ, 𝜎) will return the cumulative distribution function of a normal distribution with certain mean and SD