## **Learning Outcomes**

This article explains how major continuous distributions are defined, parameterized, and applied in financial risk management. After reading, you will be able to:

- Distinguish the Uniform, Normal, Lognormal, Student’s t, and Chi-square distributions using their parameters, support, and shape
- Compute interval probabilities and quantiles using standardization (z-scores) and distribution tables/critical values
- Select an appropriate distribution for common FRM tasks such as confidence intervals, volatility uncertainty, and parametric VaR

## **GARP FRM Part 1 Syllabus**

For the GARP FRM Part 1 exam, you are expected to understand continuous distributions and their use in risk measurement with a focus on the following syllabus points:

- Distinguish key properties and common occurrences of Uniform, Normal, Lognormal, Chi-squared, and Student’s t distributions (LO 14.a)
- Interpret PDFs, CDFs, and quantiles for continuous random variables (LO 13.d, LO 13.e)
- Relate distributional shape (skewness, kurtosis, fat tails) to financial returns and model selection, including why t tails matter (LO 23.c, LO 23.e)

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. A random variable $X$ is Uniform on $[12, 28]$. What is $P(15 \le X \le 25)$?  
   a) $0.509$  
   b) $0.625$  
   c) $0.950$  
   d) $1.000$

2. If $R \sim N(0.10, 0.12^2)$ (returns in decimals), what is the z-score of a 0.30 return?

3. In one sentence: when is a Student’s t distribution used instead of a normal distribution for inference about a mean?

4. True/False (brief explanation required): A Chi-square random variable can take negative values when the degrees of freedom are large.

## **What Do You Know Already?**

- Can you explain, in words, why $P(X=x)=0$ for a continuous random variable but $P(x_1 \le X \le x_2)$ can be positive?
- Do you remember how to convert a Normal variable to a Standard Normal and read the probability from a z-table?
- Can you state why lognormal variables are used for prices, while returns can be negative?

Use these prompts to gauge your starting point; then focus your revision where your answers were least confident.

## **Introduction**

Continuous distributions are the workhorse of quantitative risk management. Market prices, interest rates, volatilities, and portfolio P&L are often modeled as continuous random variables, so the FRM toolkit relies on probability density functions (PDFs), cumulative distribution functions (CDFs), and quantiles to translate uncertainty into decisions.

For the FRM Part 1 exam, the most frequently used continuous distribution is the normal distribution, mainly because it is analytically convenient and central to statistical testing, confidence intervals, and parametric Value-at-Risk (VaR). However, real financial returns often deviate from normality through skewness and excess kurtosis (fat tails), which motivates alternative choices such as the Student’s t distribution. Meanwhile, the lognormal distribution appears throughout derivatives pricing because it is consistent with strictly positive prices, and the chi-square distribution appears in inference about variance (volatility uncertainty).

> **Key Term: Probability density function (PDF)**  
> A PDF, f(x), describes a continuous random variable by assigning density (not point probability) to outcomes. Interval probabilities are areas under the curve:  
> $$P(a \le X \le b)=\int_a^b f(x)\,dx$$

## **Continuous distributions and probability statements**

A continuous random variable can take infinitely many values on an interval (or the real line). Because there are infinitely many distinct outcomes, the probability of any single exact value is zero:

- $P(X=x)=0$ for any specific $x$
- probabilities are computed over ranges, such as $P(x_1 \le X \le x_2)$

The CDF translates a value into a probability:
$$F(x)=P(X \le x)$$
and, for continuous variables, the PDF and CDF are linked by:
$$F(x)=\int_{-\infty}^{x} f(t)\,dt$$

Risk measurement is often quantile-based. VaR, for example, is a loss quantile: a level $L$ such that $P(\text{Loss} \le L)=q$ (e.g., $q=99\%$). Quantiles are therefore a natural bridge between distribution theory and risk limits.

> **Key Term: Quantile function**  
> The quantile function Q(p) is the inverse of the CDF. If F(x)=p, then Q(p)=x. In words, Q(99%) is the value exceeded only 1% of the time.

### Worked Example 1.1

A portfolio’s one-day loss $L$ has CDF value $F(2.0\%)=0.95$. Interpret this statement.

> **Answer:**  
> By definition, F(x)=P(L ≤ x). So F(2.0%)=0.95 means:
> $$P(L \le 2.0\%)=95\%$$
> Equivalently, the 95% loss quantile is 2.0%, so only 5% of days have losses greater than 2.0%.

## **Continuous uniform distribution**

A continuous uniform distribution is the simplest continuous model: all values in an interval are equally likely. If $X \sim \text{Uniform}(a,b)$ with $a<b$, then the probability of landing in a sub-interval is proportional to its length. For $a \le x_1 \le x_2 \le b$:
$$P(x_1 \le X \le x_2)=\frac{x_2-x_1}{b-a}$$

The uniform distribution is foundational in simulation: many pseudo-random number generators produce $\text{Uniform}(0,1)$ draws, which are then transformed into other distributions (normal, t, etc.) using inverse CDF (quantile) methods or acceptance-rejection methods.

A standard result (useful for quick checks) is:

- Mean: $E[X]=\frac{a+b}{2}$
- Variance: $\text{Var}(X)=\frac{(b-a)^2}{12}$

> **Key Term: Support**  
> The support of a distribution is the set of values that can occur with positive density/probability. For X ~ Uniform(a,b), the support is [a,b].

### Worked Example 2.1

Let $X \sim \text{Uniform}(12,28)$. Compute $P(15 \le X \le 25)$.

> **Answer:**  
> Use the interval-length rule:
> $$P(15 \le X \le 25)=\frac{25-15}{28-12}=\frac{10}{16}=0.625$$
> So the probability is 62.5%.

## **Normal distribution and standardization**

The normal distribution is symmetric, bell-shaped, and fully characterized by its mean and variance:
$$X \sim N(\mu,\sigma^2)$$
Its PDF is:
$$f(x)=\frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$

Key FRM-relevant properties include:

1. Symmetry: mean = median = mode, and $P(X \le \mu)=0.5$
2. Skewness is 0 and kurtosis is 3 (excess kurtosis is 0)
3. Linear combinations of independent normal variables are normal
4. Tails are “thin” relative to fat-tailed alternatives (large deviations are comparatively rare)

### Standard normal and z-scores

A standard normal random variable is $Z \sim N(0,1)$. Any normal variable can be standardized:
$$Z=\frac{X-\mu}{\sigma}$$
This converts questions about $X$ into questions about $Z$ that can be answered with a z-table (or calculator):
$$P(X \le x)=P\left(Z \le \frac{x-\mu}{\sigma}\right)$$

> **Key Term: Standardization (z-score)**  
> Standardization converts X ~ N(μ,σ^2) into Z ~ N(0,1) via z=(x-μ)/σ. The z-score measures “how many standard deviations from the mean.”

### Worked Example 3.1

Earnings per share (EPS) are modeled as normal with $\mu=\$6$ and $\sigma=\$2$. What is $P(\text{EPS} \ge \$9.70)$?

> **Answer:**  
> Step 1: Compute the z-score:
> $$z=\frac{9.70-6}{2}=1.85$$
> Step 2: Use the standard normal CDF. From a z-table, Φ(1.85)≈0.9678.  
> Step 3: Convert to the right-tail probability:
> $$P(\text{EPS} \ge 9.70)=1-\Phi(1.85)=1-0.9678=0.0322$$
> So the probability is about 3.22%.

### Confidence intervals and “two different meanings” in FRM work

In risk management practice, “confidence interval” is sometimes used as a prediction-style range for the next observation of a normal variable:

- 90%: $\mu \pm 1.65\sigma$
- 95%: $\mu \pm 1.96\sigma$
- 99%: $\mu \pm 2.58\sigma$

In statistical inference (covered elsewhere in Quantitative Analysis), a confidence interval for the _population mean_ uses the standard error $\sigma/\sqrt{n}$ (or $s/\sqrt{n}$ with a t critical value). Both ideas appear in FRM study materials, so you must read the question carefully: is it a range for the _next outcome_ or a range for the _unknown mean parameter_?

> **Key Term: Confidence interval**  
> A confidence interval is a probability-based range. In finance it may describe a likely range for an outcome (prediction-style) or, in inference, a likely range for an unknown parameter such as μ.

### Worked Example 3.2

A mutual fund has average annual return $\mu=10.5\%$ and standard deviation $\sigma=18\%$. If returns are approximately normal, what is the 95% interval for next year’s return?

> **Answer:**  
> For a normal outcome interval, use μ ± 1.96σ:
> $$10.5\% \pm 1.96(18\%)=10.5\% \pm 35.28\%$$
> Lower bound: 10.5%-35.28%=-24.78%  
> Upper bound: 10.5%+35.28%=45.78%  
> So a 95% interval is approximately [-24.78%, 45.78%].

### Normality and tail risk in returns

The normal model is convenient, but financial returns often show:

- nonzero skewness (asymmetry)
- excess kurtosis (kurtosis greater than 3), meaning more mass in the tails than a normal distribution

Because VaR and Expected Shortfall (ES) are tail measures, tail thickness is not a minor detail: it can materially change risk estimates. A common exam-relevant point is that fat tails make extreme outcomes more common than under the normal model; Student’s t is a leading example of a fat-tailed distribution.

### **Exam Warning**

> Many candidates mix up (1) standardizing an outcome x using z=(x-μ)/σ with (2) standardizing a sample mean using z=(x̄-μ)/(σ/√n) or t=(x̄-μ)/(s/√n).  
> Read the question: if it says “next year’s return,” it typically uses σ; if it says “confidence interval for the population mean,” it uses σ/√n (or s/√n with t).

## **Lognormal distribution and positive-valued variables**

A lognormal distribution arises when a variable is the exponential of a normal variable. If:

- $X \sim N(\mu,\sigma^2)$
- $Y=e^X$

then $Y$ is lognormally distributed, and $\ln(Y)=X$ is normal.

Key implications:

1. Support is strictly positive: $Y>0$. A lognormal variable cannot be negative.
2. Shape is right-skewed: large positive outcomes are possible, while negative values are impossible.
3. A natural fit for prices and levels (e.g., equity prices) rather than arithmetic returns.

This is why many asset pricing models (for example, the Black-Scholes-Merton framework introduced later in the FRM program) model _prices_ as lognormal while _log returns_ are normal.

Common moments (useful for quick computations) are:
$$E[Y]=e^{\mu+\frac{1}{2}\sigma^2}$$
$$\text{Var}(Y)=\left(e^{\sigma^2}-1\right)e^{2\mu+\sigma^2}$$
The median is $e^\mu$.

### Worked Example 4.1

Suppose an asset’s future price $S$ is lognormal with $\ln(S) \sim N(4.605, 0.25^2)$. Compute $P(S < \$80)$.

> **Answer:**  
> Step 1: Convert the price threshold to log space:
> $$\ln(80)\approx 4.382$$
> Step 2: Standardize in log space:
> $$z=\frac{4.382-4.605}{0.25}=\frac{-0.223}{0.25}=-0.892$$
> Step 3: Read the probability from the standard normal CDF:
> $$P(S<80)=P(Z<-0.892)\approx 0.187$$
> So there is about an 18.7% chance the price is below \$80.

## **Student’s t distribution and fat tails**

Student’s t distribution is similar in shape to the normal distribution but has fatter tails, meaning it assigns more probability to large deviations from the mean. It is:

- symmetric
- defined by a single parameter: degrees of freedom (df)
- closer to the normal distribution as df increases

In classical inference, t is the appropriate distribution for confidence intervals and hypothesis tests about a population mean when:

- the population variance is unknown, and
- the data are from a normal or approximately normal population, especially for small samples (often noted as $n<30$)

For inference about a mean, the statistic is:
$$t=\frac{\bar{X}-\mu}{s/\sqrt{n}}$$
with degrees of freedom $df=n-1$.

> **Key Term: Degrees of freedom**  
> Degrees of freedom (df) count how many independent pieces of information remain after estimating parameters. For a sample mean, df=n-1 because one constraint is imposed by estimating X̄.

### Worked Example 5.1

A risk analyst has $n=16$ daily returns (in %) with sample mean $\bar{r}=1.2\%$ and sample standard deviation $s=4.0\%$. Construct a 95% confidence interval for the population mean return (assume returns are approximately normal and variance is unknown).

> **Answer:**  
> Step 1: Identify degrees of freedom:
> $$df=n-1=15$$
> Step 2: Use the 95% two-sided t critical value. For df=15:
> $$t_{0.025,15}\approx 2.131$$
> Step 3: Compute the standard error:
> $$\text{SE}=\frac{s}{\sqrt{n}}=\frac{4.0\%}{\sqrt{16}}=1.0\%$$
> Step 4: Form the interval:
> $$\bar{r} \pm t_{0.025,15}\cdot \text{SE} = 1.2\% \pm 2.131(1.0\%)$$
> Lower bound: 1.2%-2.131%=-0.931%  
> Upper bound: 1.2%+2.131%=3.331%  
> The 95% confidence interval is approximately [-0.93%, 3.33%].

In market risk, the same “fatter tail” idea also motivates modeling returns with a t distribution rather than a normal distribution when calibrating tail-based measures (VaR/ES). Even if you do not compute t-based VaR in Part 1, you should understand the direction of the effect: fatter tails imply larger loss quantiles for the same volatility.

## **Chi-square distribution and volatility uncertainty**

The chi-square distribution appears whenever you sum squared standard normals. If $Z_1,\dots,Z_\nu$ are independent and each $Z_i \sim N(0,1)$, then:
$$\sum_{i=1}^{\nu} Z_i^2 \sim \chi^2_{\nu}$$

Key properties tested in FRM-style questions:

- Support: bounded below by 0
- Shape: right-skewed (asymmetric), especially for small df
- Parameter: degrees of freedom $\nu$
- As $\nu$ increases, the shape becomes closer to normal

A core application is inference about variance. If a population is normal and you sample $n$ observations, then the sample variance $S^2$ satisfies:
$$\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$$
This result is the basis for confidence intervals for the true variance $\sigma^2$ (and therefore volatility $\sigma$), which matters directly in VaR, option Greeks, and risk limits that depend on estimated volatility.

### Worked Example 6.1

You have $n=11$ monthly returns from an approximately normal process with sample variance $s^2=0.0225$ (so sample volatility is 15%). Construct a 95% confidence interval for the true variance $\sigma^2$. Use $df=10$ critical values: $\chi^2_{0.975,10}=20.483$ and $\chi^2_{0.025,10}=3.247$.

> **Answer:**  
> Step 1: Use the chi-square interval form:
> $$P\left(\chi^2_{0.025,df} \le \frac{(n-1)s^2}{\sigma^2} \le \chi^2_{0.975,df}\right)=0.95$$
> Rearranging gives the variance interval:
> $$\frac{(n-1)s^2}{\chi^2_{0.975,df}} \le \sigma^2 \le \frac{(n-1)s^2}{\chi^2_{0.025,df}}$$
> Step 2: Plug in values. Here (n-1)=10:
> $$\sigma^2_{\text{low}}=\frac{10(0.0225)}{20.483}=0.01098$$
> $$\sigma^2_{\text{high}}=\frac{10(0.0225)}{3.247}=0.06930$$
> The 95% confidence interval for variance is approximately [0.0110, 0.0693].  
> Step 3 (optional): Convert to volatility by square root:
> $$\sigma_{\text{low}}\approx \sqrt{0.01098}=10.5\%$$
> $$\sigma_{\text{high}}\approx \sqrt{0.06930}=26.3\%$$

This example shows why volatility estimation is itself a risk: even with a normal model, limited sample size can produce a wide confidence interval for $\sigma$.

## **Summary**

Uniform, normal, lognormal, Student’s t, and chi-square distributions form a core set of continuous models used throughout FRM Part 1. The uniform distribution supports simulation and “equal likelihood” modeling on an interval. The normal distribution supports standardization, z-tables, and many closed-form calculations, including outcome intervals and (under additional assumptions) parametric VaR.

Lognormal modeling follows naturally when a variable must stay positive, which is why prices (not arithmetic returns) are often treated as lognormal. Student’s t is central when variance is unknown in small samples and is also a practical model for fat-tailed financial returns. Chi-square connects directly to squared normal quantities and is the key distribution for variance (volatility) inference.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Continuous random variables assign probability over intervals, computed as areas under a PDF
- Uniform($a,b$) probabilities depend only on interval length: $(x_2-x_1)/(b-a)$
- Normal($\mu,\sigma^2$) variables are standardized using $z=(x-\mu)/\sigma$ to use z-tables
- Normal distributions have skewness 0 and kurtosis 3; financial returns often have excess kurtosis (fat tails)
- Lognormal variables are strictly positive and arise when $Y=e^X$ with $X$ normal
- Student’s t has fatter tails than normal and uses $df=n-1$ for inference about means with unknown variance
- Chi-square is right-skewed, bounded below by zero, and used for variance inference via $(n-1)S^2/\sigma^2 \sim \chi^2_{n-1}$

## **Key Terms and Concepts**

- Probability density function (PDF)
- Cumulative distribution function (CDF)
- Quantile function
- Support
- Continuous uniform distribution
- Normal distribution
- Standard normal distribution
- Standardization (z-score)
- Confidence interval
- Lognormal distribution
- Student’s t distribution
- Degrees of freedom
- Chi-square distribution


Probability, random variables, and distributions - Continuous distributions (Uniform, Normal, Lognormal, t, Chi-square)