Normal Distribution

A random variable is said to have a normal distribution (Gaussian curve) if its values make a smooth curve that assumes a “bell shape.” A normal variable has a mean \(μ\), pronounced as “mu,” and a standard deviation \(σ\), pronounced as “sigma.” All normal distributions have a distinguishable bell shape regardless of the mean, variance, and standard deviation.

A normal distribution has certain properties that make it a useful tool in the world of finance.

1. Its shorthand notation is \(X \sim N(\mu, \sigma^2)\). This is read as “the random variable X has a normal distribution with mean \(μ\) and variance \(σ^2\).”
2. It has a symmetric shape, meaning it can be cut into two halves that are mirror images of each other; as such, skewness = 0.
3. Kurtosis = 3. Remember that kurtosis is a measure of flatness and excess kurtosis is measured relative to 3, the “normal kurtosis.”
4. The mean, mode, and median are all equal and lie directly in the middle of the distribution.
5. If \(X \sim N(\mu_x, \sigma^2_x)\) and \(Y\sim N(\mu_y, \sigma^2_y)\), then the two variables combined also have a normal distribution.
6. The standard deviation measures the distance from the mean to the point of inflection, which is the point where the curve changes from an “upside-down-bowl” shape to a “right-side-up-bowl” shape.
7. Probabilities follow the empirical rule. About 68% of the total values lie within one standard deviation of the mean, 95% of the values lie within two standard deviations of the mean, and 99.7% lie within three standard deviations of the mean.

Area Under the Normal Distribution Curve

To determine the probability that a random variable \(X\) lies between two points \(a\) and \(b\):

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

The normal distribution is very important in statistical analysis, especially because of the central limit theorem. The theorem asserts that any distribution becomes normally distributed when the number of variables is sufficiently large. For instance, the binomial distribution tends to “change” into the normal distribution with mean \(nθ\) and variance \(nθ (1 – θ)\).

Question

Which of the following is least likely a property of normal distribution?

1. The excess kurtosis is equal to 3.
2. The mean, mode, and median are all equal.
3. About 68% of the total values lie within one standard deviation of the mean.

Solution

The correct answer is A.

The normal distribution has an excess kurtosis of 0. Excess kurtosis is a value above kurtosis of 3. Since the kurtosis of the normal distribution is 3, the excess kurtosis of 0.

B and C are incorrect. They are the true properties of the normal distribution.