2.7 Applications of normal (Gaussian) distribution

  • A wide range of continuous random variables follow a normal probability distribution

  • Continuous random variable is a variable that can take any value in a given interval, theoretically it has an infinite range from \(-\infty\) to \(+\infty\)

  • Normal distribution is a bell shaped and symmetrical with location parameter \(\mu\) and variation (dispersion) parameter \(\sigma\)

\[\begin{equation} X \sim N(\mu,~\sigma) \end{equation}\]

  • By varying the parameters \(\mu\) (population mean) and \(\sigma\) (population standard deviation) we obtain different normal distributions according to its probability density function \(f(x)\)

\[\begin{equation} f(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{1}{2}\bigg( \frac{x-\mu}{\sigma} \bigg)^2} \tag{2.12} \end{equation}\]

Three normal distrubutions with different means but the same standard deviation

FIGURE 2.1: Three normal distrubutions with different means but the same standard deviation

  • Normal distribution (with any mean and any variance/standard deviation) can be transformed into the standardized normal distribution \(Z\) with zero mean and unit variance

\[\begin{align} z&=\frac{x-\mu}{\sigma} \\ \\ Z& \sim N(0,~1) \\ \tag{2.13} \end{align}\]

  • The problem of finding the area under the curve within a certain interval of variable \(X\) is transformed into finding the area under the curve of standardized units \(Z\)

  • To compute probabilities \(P(X \leq x)\) from normal distribution in Excel you can use function =NORM.DIST(x;mean;st.dev;TRUE) with setting the cumulative distribution function to TRUE (last argument of the function)

  • To compute probabilities \(P(Z \leq z)\) from standardized normal distribution in Excel you can use function =NORM.S.DIST(z;TRUE) with setting the cumulative distribution function to TRUE

Example 2.10 Suppose \(X\) is normal random variable with mean \(8\) and standard deviation \(5\). Find the probability \(P(X \leq 8.6)\).

\(P(X \leq 8.6)=\)NORM.DIST(8,6;8;5;TRUE)

\(P(Z \leq 0.12)=\)NORM.S.DIST(0,12;TRUE)

Example 2.11 Suppose that the average number of hours a personal computer is used for entertainment is two hours per day. Assume the times for entertainment are normally distribued and the standard deviation is half an hour.

  1. Find the probability that a personal computer is used for entertainment more than 1 hour per day

  2. Find the probability that a personal computer is used for entertainment more than 4 hours per day

  3. Find the probability that a personal computer is used for entertainment between 1 and 4 hours per day

  • Variance of returns is just one possible risk measure, i.e.¬†different risk measures can be employed when dealing with returns of risky asset
Value-at-Risk is very popular in practice as it measures the maximum possible loss with a given
probability (on the left tail) during a certain time period
  • A normal distribution of returns is usually assumed when computing VaR at probability level of \(5\%\)

\[\begin{equation} VaR(5\%)=\mu-1.96 \cdot \sigma \tag{2.14} \end{equation}\]

Example 2.12 Consider two stocks, \(X\) and \(Y\). Returns of stock \(X\) are normally distributed with mean \(1\%\) and variance \(4\%\). Returns of stock \(Y\) are normally distributed with mean \(5\%\) and variance \(16\%\). The stock returns have a positive correlation \(0.7\). Compute portfolio Value-at-Risk at \(5\%\) probability level if portions of both stocks are equal.