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Cumulative Distribution Function

It is a function that describes the probability distribution of a random variable by specifying the probability that the random variable takes on a value less than or equal to a given value.

The CDF of a random variable \(X\) , denoted \(F_X(X)\) , is a function from \(\mathbb{R} \to [0,1]\) is defined as

\[F_X(X) = P(X \leq x)\]

Properties

  • \(F(X)\) is always a non-decreasing funciton taking values between 0 and 1.
  • \(P(a < X \leq b) = F_X(b) - F_X(a)\)
  • As \(X \to - \infty\) , \(F_X\) goes to 0.
  • As \(X \to \infty\) , \(F_X\) goes to 1.
  • Probability of \(X\) taking a specific value is always 0.

CDF Of Standardised Variables

Let a discrete random variable \(X\) have a CDF \(F_X\). Assume that \(Y = \frac{X - \mu}{\sigma}\) , where \(\mu\) and \(\sigma\) are the mean and standard deviation of \(X\) respectively. If \(F_Y\) is the CDF of \(Y\) , then

\[F_Y(y) = F_X(\mu + Y \sigma)\]

Continuous Random Variable

Example

Height , Weight , Time , Temperature , Price , etc.

A continuous random variable is a type of random variable that can take on any value within a specified range or interval. Unlike discrete random variables, which can only assume a countable set of distinct values, continuous random variables have an infinite number of possible values within a given range.

A random variable \(X\) with CDF \(F_X(x)\) is said to be continuous random variable if \(F_X(x)\) is continuous at every \(x\).

Properties

  • \(P(X =x) = 0\) for all \(x\)
  • \(\therefore P(a \leq X \leq b) = P(a < X \leq b) = P(a \leq X < b) = P(a < X < b)\)
  • Graphs of continuous random variables never breaks at any point and does not jump from one value to another.
  • Probability of \(X\) falling in an interval will be nonzero

Probability Density Functions

Probability Density Function (PDF) is a function that describes the probability distribution of a continuous random variable.

The PDF represents the relative likelihood of different outcomes or values occurring within that range. It is typically denoted as \(f(x)\), where \(x\) is the variable, and the integral of the PDF over a specific interval gives the probability of the random variable falling within that interval.

\[\int^{b}_{a}f(x) dx = F(b) - F(a) = P(a < X < b)\]

Properties

  • The PDF must be non-negative for all values of x.

  • The total area under the PDF curve must be equal to 1.

    \(\implies \int^{\infty}_{ - \infty} f(x) dx = 1\)

Common Distribution Functions

Distribution PDF CDF
Uniform \(X \sim Uniform[a,b]\) \(f_X(x) = \begin{cases} \frac{1}{b-a} & a < x < b\\ 0 & \text{otherwise} \end{cases}\) \(F_X(x) = \begin{cases} 0 & x \leq a \\ \frac{x-a}{b-a} & a < x < b \\ 1 & x \geq b \end{cases}\)
Exponential \(X \sim Exp(\lambda)\) \(f_X(x) = \begin{cases} \lambda \exp(-\lambda x) & x>0 \\ 0 & otherwise \end{cases}\) \(F_X(x) = \begin{cases} 0 & x \leq 0 \\ 1 - \exp(-\lambda x) & x >0 \end{cases}\)
Normal \(X \sim Normal(\mu , \sigma^2)\) \(F_X(x) = \frac{1}{\sigma \sqrt{2 \pi}} \exp(- \frac{(x - \mu)^2}{2 \sigma^2})\) \(F_X(x) = \int^{x}_{- \infty} f_X(u)du\)