Limited Time Offer: Save 10% on all 2021 and 2022 Premium Study Packages with promo code: BLOG10    Select your Premium Package »

Bernoulli Random Variables and Binomial Random Variables

Bernoulli Random Variables and Binomial Random Variables

Probability distributions have different shapes and characteristics. As such, we describe a random variable based on the shape of the underlying distribution.

A Bernoulli Random Variable

A Bernoulli trial is an experiment that has only two outcomes: success (S) or failure (F). You should note that we use the words “success” and “failure” just for labeling purposes and therefore these words may not necessarily carry with them the ordinary meanings.

We define the random variable \(X\) by \(X(s) = 1\), with a probability of \(p\) and \(X(f) = 0\) with a probability of \(1-p\). A such, \(X\) is the number of successes that occur with a probability between 0 or 1. The probability function of Bernoulli random variables is given by:

$$
f(x)=\left\{\begin{array}{ll}
p^{x} \times (1-p)^{1-x} & \text { if } x=0,1 \\
0 & \text { elsewhere }
\end{array}=\left\{\begin{array}{cl}
p & \text { if } x=1 \\
1-p & \text { if } x=0
\end{array}\right.\right.
$$

A Bernoulli variable can sometimes be used as an “indicator” to indicate whether a given event occurs. We could set \(X = 1\) if event B occurs and \(X = 0\) if event B does not occur. For example, event B could be a return of over 10% on a stock.

A Binomial Random Variable

A binomial random variable is the number of successes in n Bernoulli trials where:

  • The trials are independent: the outcome of any trial does not depend on the outcomes of the other trials.
  • The trials are identical: The probability of success is equal for all trials.

For example, the tossing of a coin has two mutually exclusive outcomes, where the probability of the outcome of any toss (trial) is not affected by prior outcomes from prior trials.

The Binomial Distribution

The binomial distribution is a sequence of \(n\) Bernoulli trials where the outcome for every trial can be a success or a failure.

Suppose the probability of a success is \(p\):

$$ P\left( X=x \right) =\left( \begin{matrix} n \\ x \end{matrix} \right) { p}^{ x }\left( 1-p \right) ^{ n-x },x=0,1,2,…,n;0<p<1 $$

Where:

$$ \left( \begin{matrix} n \\ x \end{matrix} \right) ={ \quad }^{ n }{ C }_{ x }=\frac { n! }{ \left( n-x \right) !x! } $$

$$ \text{Mean of X} = n p $$

$$ \text{Variance of X} = n p (1 – p )$$

Note to candidates: Sometimes, the probability of success can be denoted by \(p\) and \((1-p)\) will denote the probability of failure.

Example: Binomial distribution

The probability of surviving an attack by a certain disease is 60%. What is the probability that at least 11 out of a group of 12 people affected by the disease will survive?

Solution

The number of survivors is distributed binomially with parameters \(n = 12\) and \(θ = 0.6\):

$$ \begin{align*} P(X \ge 11) & = P(X = 11 \text{ or } 12) \\ &={ ^{ 12 }{ C }_{ 11 }{ \times 0.6 }^{ 11 }\times { 0.4 }^{ 1 }+^{ 12 }{ C }_{ 12 }{ \times 0.6 }^{ 12 } } \\ &= \cfrac {12!}{(1!11!)} 0.6^{11} \times 0.4^1 + \cfrac {12!}{(0!12!)} \times 0.6^{12} \\ &= 12 \times  0.003628 \times  0.4 + 0.6^{12} \\ &= 0.01959 \end{align*} $$

Note to candidates: When the probability of success is 0.50, the binomial distribution is symmetric when the probability of success on a trial is 0.50. When the probability of success is not equal to 0.50, the binomial distribution is non-symmetrical or skewed.

Question

A bowl contains blue and orange balls. The probability of drawing a blue ball in any attempt is 0.5. Suppose you draw 5 balls from the bag, what is the probability that 3 of the 5 balls drawn are blue?

  1. 3.125%
  2. 25%
  3. 31.25%

Solution

The correct answer is C.

$$\begin{align}P(X = 3) &={5\choose 3} \times 0.5^3 \times 0.5^2\\&= 10 \times 0.03125\\&= 0.3125\end{align}$$

Featured Study with Us
CFA® Exam and FRM® Exam Prep Platform offered by AnalystPrep

Study Platform

Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Online Tutoring
    Our videos feature professional educators presenting in-depth explanations of all topics introduced in the curriculum.

    Video Lessons



    Sergio Torrico
    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
    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
    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
    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
    Marwan
    2021-06-22
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    2021-05-10
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    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
    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.