Normal distributions are a family of distributions that have the same general shape.

• They are symmetrical with scores more concentrated in the middle than in the tails.
• Normal distributions are sometimes described as bell-shaped with a single peak at the exact center of the distribution.
• The tails of the normal curve extends indefinitely in both directions. That is, possible outcomes of a normal distribution lie between - ∞ and + ∞.
• Normal distributions may differ in how spread-out they are.

The graph looks like this:

The key properties are:

• The normal distribution is completely described by two parameters: the mean (μ) and the standard deviation (σ).
• The normal distribution is symmetrical: it has a skewness of 0, a kurtosis (it measures the peakedness of a distribution) of 3, and an excess kurtosis (which equals kurtosis less 3) of 0. As a consequence, the mean, median, and mode are all equal for a normal random variable.
• A linear combination of two or more normal random variables is also normally distributed.

One reason the normal distribution is important is that many psychological, educational, and financial variables are distributed approximately normally. Measures of reading ability, introversion, job satisfaction, and memory are among the many psychological variables approximately normally distributed. Although the distributions are only approximately normal, they are usually quite close.

A second reason is that it is easy for mathematical statisticians to work with. Many kinds of statistical tests can be derived for normal distributions. Almost all statistical tests discussed in the textbook assume normal distributions. Fortunately, these tests work very well even if the distribution is only approximately normally distributed. Some tests work well even with very wide deviations from normality.

Finally, if the mean and standard deviation of a normal distribution are known, it is easy to convert back and forth from raw scores to percentiles.

For example, normal distribution is an approximate model for asset returns. The price of any asset can only drop to 0. Therefore, the lowest return on an asset is -100% (i.e., all investment in the asset is lost). Since the normal distribution extends to negative infinity without limit, it is not an accurate model for asset returns. However, for the normal distribution, the probability of outcomes below -100% is very small. Therefore, the normal distribution can be considered an approximate model for returns. However, the normal distribution tends to underestimate the probability of extreme returns.

Confidence Intervals for a Normally Distributed Random Variable.

Analysts can use the sample mean to estimate the population mean, and the sample standard deviation to estimate the population standard deviation. The sample mean and sample standard deviation are point estimates.

Probability statements about a random variable are often framed using confidence intervals built around point estimates. In investment work, confidence intervals for a normal random variable in relation to its estimated mean are often used.

Confidence intervals use point estimates to make probability statements about the dispersion of the outcomes of a normal distribution. A confidence interval specifies the percentage of all observations that fall in a particular interval.

The exact confidence intervals for a normal random variable X:

• 90% confidence interval for X is: x-bar - 1.645σ to x-bar + 1.645σ: this means that 10% of the observations fall outside the 90% confidence interval, with 5% on each side.

• 95% confidence interval for X is: x-bar - 1.96σ to x-bar + 1.96 σ: this means that 5% of the observations fall outside the 95% confidence interval, with 2.5% on each side.

• 99% confidence interval for X is: x-bar - 2.58 σ to x-bar + 2.58 σ: this means that 1% of the observations fall outside the 99% confidence interval, with 0.5% on each side.

Hint: memorize these numbers (1.645, 1.96 and 2.58) to quickly solve relevant problems. For details about confidence intervals, refer to Reading 5 - Sampling and Estimation.

The Univariate and Multivariate Distributions

To this point, the focus has been on distributions that involve only one variable, such as the binomial, uniform, and normal distributions. A univariate distribution describes a single random variable. For example, suppose that you would like to model the distribution of the return on an asset. Such a distribution is a univariate distribution.

A multivariate distribution specifies the probabilities for a group of related random variables. It is used to describe the probabilities of a group of continuous random variables if all of the individual variables follow a normal distribution.

Each individual normal random variable would have its own mean and its own standard deviation, and hence its own variance. When you are dealing with two or more random variables in tandem, the strength of the relationship between (or among) the variables assumes huge importance. You will recall that the strength of the relationship between two random variables is known as the correlation.

When there is a group of assets, the distribution of returns on each asset can either be modeled individually or on the assets as a group. A multivariate normal distribution for the returns on n stocks is completely defined by three lists of parameters:

• The list of the mean returns on the individual securities (n means in total).
• The list of the securities' variances of return (n variances in total).
• The list of all the distinct pairwise return correlations (n(n-1)/2 distinct correlations in total).

The higher the correlation values, the higher the variance of the overall portfolio. In general, it is better to build a portfolio of stocks whose prices are not strongly correlated with each other, as this lowers the variance of the overall portfolio.

It is the correlation values that distinguish a multivariate normal distribution from a univariate normal distribution. Consider a portfolio consisting of 2 assets (n = 2). The multivariate normal distribution can be defined with 2 means, 2 variances, and 2 x (2-1)/2 = 1 correlation. If an analyst has a portfolio of 100 securities, the multivariate normal distribution can be defined with 100 means, 100 variances, and 100 x (100 - 1)/2 = 4950 correlations. Portfolio return is a weighted average of the returns on the 100 securities. A weighted average is a linear combination. Thus, portfolio return is normally distributed if the individual security returns are (joint) normally distributed. In order to specify the normal distribution for portfolio return, analysts need means, variances, and the distinct pairwise correlations of the component securities.

Practice Question 1

For x, a normal random variable, which of the following is (are) false?

I. The parameters are N and p.
II. The parameters are the mean and standard deviation.
III. The graph is a bell-shaped curve.
IV. The probability x is equal to a particular value (say, 55) is zero.

Correct Answer: I only

N and p are parameters for a binomial random variable.

Practice Question 2

For x, a normal random variable, which of the following is (are) false?

I. P(x < mean) = 50%
II. P(x < 38) = P(x < 38)
III. The higher the standard deviation the more spread out the distribution.
IV. P(35 < x < 50) = P(x <50) - P(x < 35)

Correct Answer: I

P(x < mean) = 50%. A normal distribution is centered (symmetrical about) the mean. The area under the curve to the left of the mean, P(x <= mean), is 0.5. The area under the curve to the right of the mean, P(x >= mean), is 0.5. The total area under the bell-shaped curve is 1.

Practice Question 3

For x, a normal random variable, which of the following is false?

A. P(Q1 < x < Q3) = 50%
B. Q1 is 0.67 standard deviations below the mean.
C. x-score that cuts off the top 10% of the distribution is 1.28 standard deviations below the mean.

Correct Answer: C

The x-score that cuts off the top 10% of the distribution must be above the mean. In fact, the x-score that cuts off the top 10% of the distribution is 1.28 standard deviations above the mean.

Practice Question 4

Which of the following is (are) false regarding the normal curve?

I. The smaller the standard deviation, the steeper the bell curve.
II. The mean locates the axis of symmetry.
III. Total area under the curve is one, with 0.5 area to either side of the mean.
IV. The mean is greater than the mode.

Correct Answer: IV

Practice Question 5

Which of the following is not a characteristic of the normal distribution?

A. unimodal
B. symmetrical about the median
C. extends from 0 to infinite

Correct Answer: C

The range of the normal distribution is from -infinite to infinite, not from 0 to infinite.

Practice Question 6

Which of the following statements about a normal distribution is the LEAST ACCURATE? A normal distribution ______.

A. has an excess kurtosis of 3
B. is completely described by two parameters
C. can be the linear combination of two or more normal random variables

Correct Answer: A

A normal distribution has a kurtosis of 3. Its excess kurtosis (kurtosis - 3.0) equals zero.

Practice Question 7

Which of the following statements is untrue?

A. The probabilities for a group of related variable can be described using a multivariate distribution.
B. A univariate distribution has a variance of 1.
C. A univariate distribution always describes the probability of a single variable.
D. Multivariate distributions are commonly used in investment analysis.

Correct Answer: B

This statement is incorrect because a univariate distribution refers to a distribution of a single variable; it says nothing about the variance of the distribution. All the other statements are correct.

Practice Question 8

A multivariate distribution is sometimes called ______.

A. multiple distribution
B. likelihood
C. multidistribution

Correct Answer: B

A multivariate distribution is also called likelihood or joint distribution.

Practice Question 9

Which of the following statements is not correct?

A. Given the covariances and the variances, one can sometimes find the correlations.
B. Given the correlations and the variances, one can obtain the covariances.
C. Given the standard errors and the correlation, one can obtain the covariances.
D. Given the correlations and the covariances, one can obtain the variances.

Correct Answer: A

Given the variances and the covariances, we can always (not sometimes) obtain the correlations.

Practice Question 10

Which of the following does (do) not describe a multivariate normal distribution for the return on n stocks?

I. n mean returns
II. n variances
III. sum of n mean returns
IV. n(n-1)/2 correlations

Correct Answer: III only

A multivariate normal distribution is completely specified if one has the lists for the mean returns, variances, and correlations or covariances between the variables.

Practice Question 11

Of the following statements:

I. In a normal distribution, the coefficient of skewness is 1.0.
II. In a normal distribution, approximately 95% of the observations lie within one standard deviation of its mean.

A. Only I is true.
B. Only II is true.
C. Neither I nor II is true.

Correct Answer: C

Practice Question 12

Which of the following is (are) true about a normal distribution?

I. It is a bimodal distribution.
II. It can be characterized completely by a single parameter.
III. It ranges from negative infinity to positive infinity.
IV. It is positively skewed.

A. III and IV
B. III only
C. II and III

Correct Answer: B

A normal distribution is completely characterized by 2 parameters, mean and variance. It ranges from negative infinity to positive infinity, has a single mode or peak (occurring at the mean), is symmetrical about the mean, and has zero skewness.

Practice Question 13

The time it takes an auto mechanic to replace a carburetor is known to follow a normal distribution with a mean of 53 minutes and a standard deviation of 7.5 minutes. This means that ______

A. the peak of the bell curve is at 53.
B. the peak of the bell curve is at 7.5.
C. it always takes at least 53 minutes to replace a carburetor.

Correct Answer: A

The peak of the bell-curve occurs at the mean of 53.

Practice Question 14

The time it takes an auto mechanic to replace a carburetor is known to follow a normal distribution with a mean of 53 minutes and a standard deviation of 7.5 minutes. This means that ______

A. about 68% of the time it takes between 45.5 minutes and 53 minutes to replace a carburetor.
B. about 68% of the time it takes between 53 minutes and 60.5 minutes to replace a carburetor.
C. about 95% of the time it takes between 38 minutes and 68 minutes to replace a carburetor.

Correct Answer: C

95% of the observations lie within μ +- 2 σ or (38,68).

Practice Question 15

Two normal distributions, X and Y, have means of 10% and 15% respectively. What is the mean of the following linear combination: 2X + 3Y?

A. 62.5%
B. 65%
C. 31.3%

Correct Answer: B

The linear combination of two normal distributions is also a normal distribution. Its mean = 2 x 10% + 3 x 15% = 65%.

Practice Question 16

A multivariate distribution ______.

A. specifies the probabilities for a group of related random variables
B. describes a single random variable
C. helps investors make risk-averse decisions

Correct Answer: A

A common multivariate distribution is the multivariate normal distribution, which is a "combination" of a number of univariate normal random variables. It is important to note that when each individual random variable is normally distributed, the multivariate distribution that comprises these individual random variables is normal as well.