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

The correct answer is B.

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.