# Probability Distribution of Discrete and Continous Random Variables

## Probability Distribution

The probability distribution of a random variable $$X$$ is a graphical presentation of the probabilities associated with the possible outcomes of $$X$$. A random variable is any quantity for which more than one value is possible. The price of quoted stocks is a good example in this respect. Simply put, a probability distribution gathers all the outcomes and further indicates the probability associated with each outcome.

This reading covers the probability distributions listed below:

• Uniform.
• Binomial.
• Normal.
• Lognormal.
• Student’s t.
• Chi-square.
• F-distribution.

#### Example: Probability Distribution

Suppose we roll a die. The set of possible outcomes is:

{ 1   2   3   4   5   6 }

Each of these outcomes would occur with a probability of 1/6 because each outcome has an equal chance of occurrence. Consequently, the probability distribution would be a straight line:

Note to candidates: Although the above distribution is a straight line, most real-life distributions are usually curved. We will particularly delve into the bell-shaped normal distribution later.

## Discrete Random Variables

A discrete random variable can take on a finite number of outcomes. Examples include:

1. If we roll a die, there are 6 possible outcomes. Therefore, the outcomes are discrete and random.
2. The number of CFA® charterholders employed within a given year is a discrete random variable.

## Continuous Random Variables

A continuous random variable is one that has an infinite number of possible outcomes. A good example can be the rate of return on a stock. For instance, the return can be 6%, or between 6% and 7%, in which case, it can take on 6.4%, 6.41%, 6.412%, or even 6.412325%, i.e., infinite values.

## Probability Function

A probability function gives the probability of a random variable $$X$$ taking on a value “$$x$$.” The probability functions of discrete and continuous random variables are slightly different.

For a discrete random variable, the probability function, $$P(x)$$, satisfies the following properties:

1. $$P(X = x) = P(x)$$.
In statistics, $$P(x)$$ is said to be a probability mass function.
2. $$P(x)$$ is always nonnegative for all $$x$$.
3. The sum of the probabilities of all possible outcomes = 1.

For a continuous random variable, the probability function, $$f(x)$$, satisfies the following properties:

1. The probability that $$x$$ is between two values, $$a$$ and $$b$$ is:
$$P\left\{ a\le x\le b \right\} =\int _{ a }^{ b }{ f\left( x \right) dx }$$
$$f(x)$$ is said to be a probability density function (pdf).
2. $$f(x)$$ is nonnegative for all $$x$$.
3. The integral of the probability function is 1, that is,
$$\int _{ -\infty }^{ \infty }{ f\left( x \right) dx=1 }$$

### Question

Which of the following is least likely a property of probability mass function, P(x)?

1. The probability mass function is nonnegative for all $$x$$.
2. The integral of the probability mass function is 1.
3. The sum of the probabilities of all possible outcomes = 1.

Solution

Option B is a property of probability density function (for continuous random variables) and not probability mass function.

A and C are incorrect. For a discrete random variable, the probability function is termed as probability mass function with the following properties:

1. $$P(X = x) = P(x)$$.
In statistics, $$P(x)$$ is said to be a probability mass function.
2. $$P(x)$$ is always nonnegative for all $$x$$.
3. The sum of the probabilities of all possible outcomes = 1.
Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success

Sergio Torrico
2021-07-23
Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
diana
2021-07-17
So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
Kriti Dhawan
2021-07-16
A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
nikhil kumar
2021-06-28
Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
Marwan
2021-06-22
Great support throughout the course by the team, did not feel neglected
Benjamin anonymous
2021-05-10
I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
Daniel Glyn
2021-03-24
I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
michael walshe
2021-03-18
Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.