9.2: Normal Distributions

Question 1

What are the 5 properties of normal distribution?

Answer 1

1. Normal distribution is described by its mean and variance, stated as X ~ N(mean, variance).
2. Skewness = 0 such that the normal distribution is symmetrical about its mean so that mean = median = mode.
3. Kurtosis = 3 which is a measure of how flat the distribution is.
4. A linear combination of normally distributed random variables is also normally distributed.
5. The probabilities of outcomes further above and below the mean get smaller and smaller but will not reach 0 (tails only get thinner but extend indefinitely).

Question 2

What is univariate distribution?

Answer 2

Univariate distribution is the distribution of a single random variable.

Question 3

What is a multivariate distribution? What 2 variables can have multivariate distributions?

Answer 3

Multivariate distribution specifies the probabilities associated with a group of random variables and is meaningful only when the behaviour of each random variable in the group is dependent upon the behaviour of the others.

Discrete and continuous random variables can have multivariate distributions. For discrete variables, multivariate distributions are described using joint probability tables. For continuous random variables, a multivariate normal distribution may be used to describe them if all of the individual variables follow a normal distribution.

Question 4

What is the parameter that is used to distinguish between a multivariate and univariate normal distribution?

Answer 4

Correlation is used to indicate the strength of the linear relationship between a pair of random variables.

Question 5

What are the parameters used to describe a univariate normal distribution? Multivariate distribution?

Answer 5

Univariate: mean and variance

Multivariate:

1. n means of the n series of returns
2. n variances of the n series of returns
3. 0.5n(n - 1) pair-wise correlation

Question 6

What is confidence interval and how is it measured? What are the 3 confidence intervals of most interest?

Answer 6

Confidence interval is a range of values around the expected outcome within which we expect the actual outcome to be some specified percentage of the time.

For instance, we would expect the random variable to be in 95% of the time given a 95% CI.

CI is measured with standard deviation such that 68% of the outcomes are within one standard deviation and 95% of the outcomes are within two standard deviations of the expected value. In practice, however, the values are unknown and are estimated as mean

1. 90% CI is mean minus 1.65s to mean plus 1.65s.
2. 95% CI is 1.96s
3. 99% CI is 2.58s

Question 7

What is the standard normal distribution? What value is used to represent the standardization?

Answer 7

The standard normal distribution is a normal distribution that has been standardized so that it has a mean of 0 and a standard deviation of 1.

Z-value represents the number of standard deviations a given observation is from the mean population.

Question 8

Explain the process of standardization. What formula is used to standardize a random variable?

Answer 8

Standardization is the process of converting an observed value for a random variable to its z-value.

z = observation - population mean / standard deviation