The Normal Distribution

The central limit theorem states that the sum (and mean) of a large number of independent random variables is approximately normally distributed.

The normal distribution is completely described by two parameters—its mean, μ, and variance, σ², indicated as X ~ N(μ, σ²).

The normal distribution has a skewness of 0 (it is symmetric), a kurtosis of 3; excess kurtosis 0. The mean, median, and the mode are all equal.

A linear combination of two or more normal random variables is also normally distributed.

The normal density with μ = 0 and σ = 1 is called the standard normal distribution (or unit normal distribution).

Approximately 68% of all observations fall in the interval μ ± σ.

Approximately 90% fall in μ ± 1.65σ.

Approximately 95% fall in μ ± 1.96σ.

Approximately 99% fall in μ ± 2.58σ.

N(x) is a conventional notation for the cdf of a standard normal variable.

P(Z ≤ 1.282) = N(1.282) = 0.90 or 90 percent

P(Z ≤ 1.65) = N(1.65) = 0.95 or 95 percent

P(Z ≤ 2.327) = N(2.327) = 0.99 or 99 percent

P(Z ≥ x) = 1.0 − N(x)

N(−x) = 1.0 − N(x)

P(Z ≥ −x) = N(x)

[Practice Problems] A portfolio has an expected mean return of 8% and standard deviation of 14%. The probability that its return falls between 8 and 11 percent is closest to:

A.  8.3%

B.  14.8%.

C.  58.3%.

[Solutions] A

P(8% ≤ Portfolio return ≤ 11%) = N(Z corresponding to 11%) – N(Z corresponding to 8%). For the first term, Z = (11% – 8%)/14% = 0.21 approximately, N(0.21) = 0.5832. 8 percent is the mean, N(Z corresponding to 8%) must equal 50%. So P(8% ≤ Portfolio return ≤ 11%) = 0.5832 – 0.50 = 0.0832 or approximately 8.3 percent.