Stats 1 week 9

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Random Variables and Their Applications

What is a Random Variable?

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. There are two main types of random variables:

• Discrete Random Variables: These can take on a countable number of distinct values. Examples include the number of heads in a series of coin flips or the number of students in a class.
• Continuous Random Variables: These can take on any value within a given range. Examples include the height of students in a class or the time it takes to run a marathon.

Uses of Random Variables in Daily Life

Random variables are used in various fields to model uncertainty and make predictions. Here are some examples:

• Finance: Modeling stock prices and returns on investments.
• Engineering: Assessing the reliability of systems and components.
• Medicine: Evaluating the effectiveness of treatments and drugs.
• Retail: Predicting the number of customers or sales in a given period.

Difference Between Discrete and Continuous Random Variables

The main difference between discrete and continuous random variables is the type of values they can take:

• Discrete Random Variables: Can take specific, countable values (e.g., 0, 1, 2, 3).
• Continuous Random Variables: Can take any value within a range (e.g., 1.5, 2.3, 3.7).

Discrete Random Variables

A discrete random variable is one that can take on a finite or countably infinite number of values. Examples include the number of defective items in a batch or the number of goals scored in a match.

Probability Mass Function (PMF)

The probability mass function (PMF) of a discrete random variable gives the probability that the variable takes on a specific value. The PMF must satisfy two conditions:

• All probabilities are non-negative.
• The sum of all probabilities is 1.

Skewed Distributions Histogram

Symmetric Distribution

Positively Skewed Distribution

Negatively Skewed Distribution

In case of symmetric(no skew) mode=median=mean.

In case of positive skewed mode<median<mean

In case of negatively skewed mode>median>mean

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable gives the probability that the variable takes on a value less than or equal to a specific value. The CDF is different from the PMF in that it accumulates probabilities up to a certain point.

Example of PMF and CDF

Consider a discrete random variable X that represents the number of heads in two coin flips. The PMF and CDF can be represented as follows:

X	PMF (P(X=x))	CDF (P(X≤x))
0	0.25	0.25
1	0.50	0.75
2	0.25	1.00