Discrete Random Variables and Variance
Discrete random variables are those that can take on a countable number of distinct values.
Variance of Discrete Random Variables
The variance of a discrete random variable X, denoted as Var(X), measures the spread of the values around the mean. It is defined as:
$$ Var(X) = E[(X - \mu)^2] $$
where $\mu = E(X)$ is the expected value or mean of X.
An alternative and often more convenient formula for calculating variance is:
$$ Var(X) = E(X^2) - [E(X)]^2 $$
This formula is particularly useful in practice as it often simplifies calculations.
Continuous Random Variables
Continuous random variables can take on any value within a given range. They are described by probability density functions (PDFs). In other words, the probability that $P(X=x) =f(x)$.
Probability Density Functions
A probability density function f(x) for a continuous random variable X has the following properties:
- $f(x) \geq 0$ for all x
- The total area under the curve of f(x) equals 1: $$ \int_{-\infty}^{\infty} f(x) dx = 1 $$
Unlike discrete probability mass functions, f(x) can take values greater than 1, as long as the total area under the curve is 1.
Piecewise Functions
PDFs can be defined piecewise, meaning different functions apply to different intervals of x.
A simple piecewise PDF might look like:
$f(x) = \begin{cases} 2x & \text{ for } 0 \leq x < 1 \\ 2-2x & \text{ for } 1 \leq x < 2 \ 0 & \text{otherwise} \end{cases}$
Measures of Central Tendency for Continuous Random Variables
Mode
The mode of a continuous random variable is the value at which the PDF reaches its maximum, which can be solved with differentiation.
Median
The median $m$ of a continuous random variable is defined as the value that divides the area under probability distribution into two equal halves:
$$ \int_{-\infty}^m f(x) dx = \frac{1}{2} $$
Mean, Variance, and Standard Deviation
These measures apply to both discrete and continuous random variables, but their calculation methods differ.
Mean (Expected Value)
For a continuous random variable X with PDF f(x):
$$ E(X) = \int_{-\infty}^{\infty} x f(x) dx $$
Which is analogous to a discrete probability distribution :
$$E(X)= \sum_k=0^\infty x_kP(X=x_k) $$
Variance
For continuous random variables, the same formula for discrete random variables apply:
$$ Var(X) = E(X^2) - [E(X)]^2 = \int_{-\infty}^{\infty} x^2 f(x) dx - \left[\int_{-\infty}^{\infty} x f(x) dx\right]^2 $$
Standard Deviation
The standard deviation is the square root of the variance:
$$ \sigma = \sqrt{Var(X)} $$
Linear Transformations of Random Variables
Linear transformations of random variables are common in statistical analysis. For a random variable X and constants a and b:
Effect on Expected Value
$$ E(aX + b) = aE(X) + b $$
Effect on Variance
$$ Var(aX + b) = a^2 Var(X) $$
Notation and Symbols
In probability theory and statistics, certain notations are commonly used:
- E(X): Expected value of X
- E(X^2): Expected value of X squared
- Var(X): Variance of X
