Probabilities

After completing this reading, you should be able to:

  • Describe and distinguish between continuous and discrete random variables.
  • Define and distinguish between the probability density function, the cumulative distribution function, and the inverse cumulative distribution function.
  • Calculate the probability of an event given a discrete probability function.
  • Distinguish between independent and mutually exclusive events.
  • Define joint probability, describe a probability matrix, and calculate joint probabilities using probability matrices.
  • Define and calculate a conditional probability, and distinguish between conditional and unconditional probabilities.

What’s a Random Variable?

A random variable is a variable whose possible values are outcomes of a random phenomenon. Examples include:

  • Flipping a coin;
  • Throwing a die; or
  • A claim on an insurance policy.

Discrete vs. Continuous Random Variables

Discrete Random Variables

A random variable X is said to be discrete:

I.If the range of all possible values is a finite set,

                e.g. {1,2,3,4,5,6}                  in the case of a six-sided die

                     or,

II. If the range of all possible values is a countably infinite set:

                e.g. {1,2,3, . . .}

Examples:

  • Picking a random stock from the S&P 500
  • The number of candidates registered for the FRM level 1 exam at any given time

Continuous Random Variables

A continuous random variable can assume any value along a given interval of a number line.

Examples:

  • Price of a stock
  • Amount time a bond sells at, say, a premium to the par value 
  • The return on a stock 

The following relationship holds for a continuous random variable \(X\):

$$ P\left[ { r }_{ 1 }<X<{ r }_{ 2 } \right] =p $$

This implies that \(p\) is the likelihood that the random variable \(X\) falls between \({ r }_{ 1 }\) and \({ r }_{ 2 }\).

The Probability Density Function, Cumulative Distribution Function, and The Inverse Cumulative Distribution Function

Probability Density Functions (PDF)

A probability density function (PDF) allows us to calculate the probability of an event.

Given a PDF f(x), we can determine the probability that x falls between a and b:

\(Pr(a<x\le b)=\int _{ a }^{ b }{ f(x)dx } \)

The density function, \(f\left( x \right) \), can be defined as follows, where \(p\) is the probability that the random variable \(X\) will be between \({ r }_{ 1 }\) and \({ r }_{ 2 }\):

$$ \int _{ { r }_{ 1 } }^{ { r }_{ 2 } }{ f\left( x \right)dx=p } $$

The probability that X lies between two values is the area under the density function graph between the two values:

Probability distribution function is another term used to refer to the probability density function. The sum of all probabilities must be equal to 1, just like in discrete random variables.

$$ \int _{ { r }_{ min } }^{ { r }_{ max } }{ f\left( x \right) dx=1 } $$

The upper and lower bounds of \(f\left( x \right) \) are defined by \({ r }_{ min }\) and \({ r }_{ max }\).

Example: Rolling a dice, and looking at the probability of every possible outcome

Probability Density Function
Probability Density Function

Cumulative Distribution Functions (CDF)

It is also called the cumulative density function and is closely related to the concept of a PDF. The likelihood of a random variable falling below a certain value is defined by a CDF. To determine the CDF, the PDF is integrated from its lower bound.

The corresponding density function’s capital letter has traditionally been used to denote the CDF. The following computation depicts a CDF, \(F\left( x \right) \), of a random variable \(X\) whose PDF is \(f\left( x \right) \):

$$ F\left( a \right) =\int _{ -\infty }^{ a }{ f\left( x \right) d\left( x \right) } =P\left[ X\le a \right] $$

The region under the PDF is a depiction of the CDF. The CDF is usually non-decreasing and varies from zero to one. We must have a zero CDF at the minimum value of the PDF. The variable cannot be less than the minimum. The likelihood of the random variable being less than or equal to the maximum is 100%.

To obtain the PDF from the CDF, we have to compute the first derivative of the CDF. Therefore:

$$ f\left( x \right) =\frac { d\left( F \right) \left( x \right) }{ dx } $$

Next, we look at how to determine the probability that a random variable \(X\) will fall between some two values \(– a\) and \(b\).

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

Where \(a\) is less than \(b\).

Cumulative Distribution Function
Cumulative Distribution Function

The following relationship is also true:

$$ P\left[ X>a \right] =1-F\left( a \right) $$

Inverse Cumulative Distribution Function

The inverse cumulative distribution function, \({ F }^{ -1 }\left( p \right)\), can be defined as follows, given that \({ F }\left( a \right) \) is a CDF:

$$ { F }\left( a \right) =p\Leftrightarrow { F }^{ -1 }\left( p \right) =a:0\le p\le 1 $$

 

Independent vs. Mutually Exclusive Events

Mutually Exclusive Events

Two events, \(A\) and \(B\), are said to be mutually exclusive if the occurrence of \(A\) rules out the occurrence of \(B\), and vice versa. For example, a car cannot turn left and turn right at the same time. To determine the likelihood of any two mutually exclusive events occurring, their individual probabilities will just be summed up. The following is the statistical notation:

$$ P\left[ A\cup B \right] =P\left[ A \right] +P\left[ B \right] $$

This is the likelihood that either \(A\) or \(B\) will occur, and is only applicable to events that are mutually exclusive. The mutually exclusive events property can be extended to any number of events.

Independent vs. Mutually Exclusive Events

Two events, A and B, are said to be mutually exclusive if the occurrence of A rules out the occurrence of B, and vice versa.

For example, a car cannot turn left and turn right at the same time.

 Two events are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both.

  • For a given random variable, the probability of any of two mutually exclusive events occurring is just the sum of their individual probabilities.
  • In statistics notation, we can write: P[A Ս B] = P[A] + P[B]

Where [A Ս B] is the union of A and B. This is the probability of either A or B occurring.

This is true only with mutually exclusive events.

Probability Matrices

A probability matrix, also referred to as a probability table, is an effective tool that summarizes the various probabilities when the joint probabilities of two variables are involved.

Assume, for instance, that a company has issued both stocks and bonds. We can downgrade the bond, upgrade it, or have its rating remain constant. The market can either be outperformed or underperformed by the stock.

Consider the following figure:

$$
\begin{array}{|ccccc|}
\hline
{} & {} & Stock & {} \\ \hline
{} & {} & Outperform & Underperform \\ \hline
Bonds & Upgrade & 17\% & 5\% & 22\% \\
{} & No\quad Change & 29\% & 24\% & 53\% \\
{} & Downgrade & 6\% & 19\% & 25\% \\
{} & {} & 52\% & 48\% & 100\% \\ \hline
\end{array}
$$

In the above figure, there is 17% chance the the stock will outperform and the bond will be upgraded. Furthermore, the likelihood that the rating of the bond will not have changed and the stock underperforms the market is 24%. To determine the unconditional probabilities, the entries can either be added across a row or down a column. Therefore, the likelihood that a stock will be upgraded no matter how it performs can be computed as:

$$ 17\%+5\%=22\% $$

In a similar streak, there is a 52% probability that the market will be outperformed by the equity, i.e.,

$$ 17\%+29\%+6\%=52\% $$

Furthermore, the sum of all the joint probabilities must be 100%.

Conditional Probability

Conditional probability is the probability of one event occurring with some relationship to one or more other events.

Example 1

Event A is that GDP will grow, and there’s a probability of 40% that this will happen.

Event B is that interest rates will rise, and that has a probability of 0.5 (50%).

A conditional probability would look at these two events in relations

Conditional Probabilities
Conditional Probabilities

hip with one another, such as the probability that there will be GDP growth given that interest rates rise.

The formula for conditional probability is:
                                    

                             \(P(A|B)=\frac { P(A\cap B) }{ P(B) } \)

Example 2

In a group of 100 investors,

  • 40 buy stocks,
  • 30 purchase bonds, and
  • 20 purchase stocks and bonds

If an investor chosen at random bought bonds, what is the probability they also bought stocks?

Event Notation Probability
Buys stocks A 0.4 (=40/100)
Buys bonds B 0.3 (=30/100)
Buys stocks and bonds A and B 0.2 (=20/100)

We want the probability of an investor buying stocks given that they have already bought bonds. This is P(A | B):

$$ P\left( A|B \right) =\frac { P\left( A\cap B \right) }{ P\left( B \right) } $$

$$ =\frac { 0.2 }{ 0.3 }  = 0.67$$

Summary

$$ P\left( A|B \right) =\frac { P\left( A\cap B \right) }{ P\left( B \right) } $$

Note that we can also make the numerator the subject so that:

$$ P\left( A\cap B \right) =P\left( A|B \right) P\left( B \right) $$

For independent events, however,

$$ P\left( A|B \right) =P\left( A \right) $$

and

$$ P\left( A\cap B \right) =P\left( A\quad and\quad B \right) =P\left( A \right) P\left( B \right) $$

Given two events \(A\) and \(B\) that are not mutually exclusive, the probability that at least one of the events will occur is given by:

$$ P\left( A\quad or\quad B \right) =P\left( A \right) +P\left( B \right) – P\left( A\cap B \right) $$

For mutually exclusive events, however,

$$ P\left( A\quad or\quad B \right) =P\left( A \right) +P\left( B \right) $$

 

Question 1

The probability that the Eurozone economy will grow this year is 18%, and the probability that the European Central Bank (ECB) will loosen its monetary policy is 52%.

Assuming that the joint probability that the Eurozone economy will grow and the ECB will loosen its monetary policy is 45%, then what is the probability that either the Eurozone economy will grow or the ECB will loosen its the monetary policy?

  1. 42.12%
  2. 25%
  3. 11%
  4. 17%

The correct answer is B.

The addition rule of probability is used to solve this question:
P(E) = 0.18 (the probability that the Eurozone economy will grow is 18%)
p(M) = 0.52 (the probability that the ECB will loosen the monetary policy is 52%)
p(EM) = 0.45 (the joint probability that Eurozone economy will grow and the ECB will loosen its monetary policy is 45%)

The probability that either the Eurozone economy will grow or the central bank will loosen its the monetary policy: 
p(E or M) = p(E) + p(M) – p(EM) = 0.18 + 0.52 – 0.45 = 0.25

Question 2

A mathematician has given you the following conditional probabilities:

p(O|T) = 0.62 Conditional probability of reaching the office if the train arrives on time
p(O|T c) = 0.47 Conditional probability of reaching the office if the train does not arrive on time
p(T) = 0.65 Unconditional probability of the train arriving on time
p(O) = ? Unconditional probability of reaching the office

What is the unconditional probability of reaching the office, p(O)?

  1. 0.5675
  2. 0.4325
  3. 0.3265
  4. 0.3856

The correct answer is A.

This question can be solved using the total probability rule.

If p(T) = 0.65 (Unconditional probability of train arriving at time is 0.65), then the unconditional probability of the train not arriving on time p(T c) = 1 – p(T) = 1 – 0.65 = 0.35.

Now, we can solve for p(O) = p(O|T) * p(T) + p(O|T c) * p(T c) = 0.62 * 0.65 + 0.47 * 0.35 = 0.5675


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