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Sahithyan's S2
Sahithyan's S2 — Methods of Mathematics

Continuous Probability Distribution

Describes a continuous random variable.

Probability Density Function

Denoted by . Does not give the probability at an exact point (which is always 0 for continuous variables). Instead, gives the relative likelihood of the random variable taking on a value in a small interval around a point.

The probability of an event is the integral of the PDF over the region corresponding to the event:

Probability of all the values combined, is 1:

Mean

The integral is taken over the entire domain of the random variable

Here is the probability density function.

Variance

An equivalent computational formula is:

Covariance

Cumulative Distribution Function

Example

Consider a continuous random variable that follows a uniform distribution over the interval [0, 4], with probability density function:

The mean would be: .

The variance would be: .

Types

Uniform Distribution

A type of continuous probability distribution where all outcomes are equally likely within a specified range. Has the PDF:

The mean, variance and CDF are:

Normal distribution

Describes data that clusters around a mean value, forming a symmetric bell-shaped curve. Denoted by . Has PDF:

Follows the empirical rule.

It’s often used to model real-world phenomena like heights, test scores, or measurement errors, where most values are near the mean, and fewer occur as you move away from it.

Standard normal distribution

A special case of the normal distribution where and . Denoted by . Has PDF:

Chi-square Distribution

Distribution of the sum of the squares of independent standard normal random variables, where is the degrees of freedom. Has the PDF:

Here:

Used in tests like the Chi-square goodness-of-fit test and tests for independence in contingency tables.

Student’s t-distribution

Probability distribution of the ratio with degress of freedom.

Here:

  • and are independent