Understanding Continuous Random Variables | CFA Level I Quantitative Methods

PREREQUISITE LESSON

This lesson is a prerequisite for the course. While you won’t be directly tested on its content in the exam, it’s assumed you’ve gained this knowledge or skill during your university studies. We strongly recommend reviewing this lesson, as its content may be essential for understanding subsequent parts of the curriculum.

In this lesson, we’ll dive into continuous random variables and their distributions. We’ll cover continuous uniform distributions, and confidence intervals.

Continuous Random Variables vs. Discrete Random Variables

In our previous lesson, we learned about discrete random variables, where the number of possible outcomes can be counted. In contrast, a continuous random variable has an infinite number of possible outcomes, even within finite boundaries.

Unlike discrete random variables, continuous random variables have distribution functions drawn with continuous lines to reflect their continuous nature. The line represents the probability for each particular outcome.

Continuous Uniform Distribution

Let’s start by examining the continuous uniform distribution. This distribution describes equally likely outcomes within a defined range. The key property of any probability distribution function is that the sum of all probabilities must equal 1. For continuous distribution functions, this means the area under the line must be 1.

For a continuous uniform distribution within an interval from A to B, the probability is 1/(B – A). For outcomes outside of this interval, the probability is 0.

The cumulative function for this distribution is 0 for outcomes before A. Between A and B, the cumulative probability rises linearly from 0 to 1. The function for this interval is (x – A) / (B – A). After point B, the cumulative probability remains at 1.

Normal Distribution

Now let’s revisit the normal distribution, which we discussed earlier in the topic on statistical concepts. Here’s a quick recap of its properties:

• It’s completely described by its mean and variance.
• It has zero skewness, meaning it’s symmetrical on both sides of its center.
• Its mean, mode, and median are equal and aligned at the center of the distribution.
• It has a kurtosis equal to 3, with leptokurtic distributions having a kurtosis greater than 3 and platykurtic distributions having a kurtosis less than 3.
• About 68% of observations lie within ±1 standard deviation from the mean, 95% within ±2 standard deviations, and 99% within ±3 standard deviations.

Confidence Intervals

When applied to probability distributions, these standard deviations can give us an idea of the level of confidence that the outcome will fall within specific ranges, which we call confidence intervals. There are three key confidence intervals to remember:

1. 90% confidence interval: ±1.65 standard deviations from the expected value.
2. 95% confidence interval: ±1.96 standard deviations from the expected value.
3. 99% confidence interval: ±2.58 standard deviations from the expected value.

Using the Z-Table

We’ve been discussing the standard normal distribution, where the mean is 0 and the standard deviation is 1. It’s easy to find the cumulative probability of any interval using the z-table.

For example, if we want to find the probability of X being less than 1.65, we can use the z-table. Look for 1.6 in the first column, then find 0.05 in the header row. The intersection of the row and column gives you the probability of X being less than 1.65, which is approximately 95%.

This figure means that 95% of the observations lie below 1.65, and we can expect X to be less than 1.65 with 95% confidence in the future.

The standard normal distribution is symmetrical about zero. So, any negative value of x will have cumulative probabilities equal to 1 minus the corresponding positive value. In this case, F(-1.65) = 1 – 0.9505 = 0.0495.

This shows that the cumulative probabilities within this interval are approximately 90%, which is the confidence interval. For exam purposes, you only need to remember ±1.65 standard deviations for a 90% confidence interval.

Understanding Z-Transform

Converting Non-Standard Normal Distributions to Standard Normal

Most real-life distributions are not standard. For example, X could be the annual earnings per share (EPS) of stocks in a particular market with a mean of $2 and a standard deviation of $0.50. The distribution is still normal, but non-standard. To convert the distribution to a standard normal, we apply two transformations:

1. Shift the distribution to align the mean to zero by subtracting the mean μ.
2. Scale the standard deviation to 1 by dividing by the standard deviation σ.

This equation can be generalised to (x – μ) / σ, known as the z-transform.

Summary

In this lesson, we covered:

• Continuous uniform distribution, where the probability is uniform for a fixed interval.
• Normal distribution, where we can determine confidence intervals based on standard deviations from the mean.
• Finding the cumulative probability using the z table.
• Converting a normal distribution to a standard normal distribution.
• Determining the most suitable portfolio using Roy’s safety first ratio.
• Using the most appropriate distribution to model different variables.