Relationship between Normal Distribution and Lognormal Distribution
A variable \(X\) is said to have a lognormal distribution if \(Y = ln(X)\) is normally distributed, where “ln” denotes the natural logarithm. In other words, when the logarithms of values form a normal distribution, we say that the original…
Shortfall Risk, Safety-first Ratio, and Identification of an Optimal Portfolio Using Roy’s Safety-first Criterion
Shortfall risk refers to the probability that a portfolio will not exceed the minimum (benchmark) return that an investor has set. In other words, it is the risk that a portfolio will fall short of the level of return an…
Using the Standard Normal Distribution to Calculate Probabilities
Using the standard normal distribution table, we can confirm that a normally distributed random variable \(Z\), with a mean equal to 0 and variance equal to 1, is less than or equal to \(z\), i.e., \(P(Z ≤ z)\). However, the…
The Standard Normal Distribution
The standard normal distribution refers to a normal distribution that has been standardized such that it has a mean of 0 and a standard deviation of 1. The shorthand notation used is: $$ N \sim (0, 1) $$ Master standard…
Normal Distribution and Confidence Intervals
A confidence interval (CI) gives an “interval estimate” of an unknown population parameter, such as the mean. It gives us the probability that the parameter lies within the stated interval (range). The precision or accuracy of the estimate depends on…
Univariate Distribution, Multivariate Distribution, and Correlation
Univariate and multivariate normal distributions are very robust and useful in most statistical procedures. Understanding their form and function will help you learn a lot about most statistical routines.
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…
Bernoulli Random Variables and Binomial Random Variables
Probability distributions have different shapes and characteristics. As such, we describe a random variable based on the shape of the underlying distribution. A Bernoulli Random Variable A Bernoulli trial is an experiment that has only two outcomes: success (S) or…
Properties of Continuous Uniform Distribution
The continuous uniform distribution is such that the random variable \(X\) takes values between \(a\) (lower limit) and \(b\) (upper limit). In the field of statistics, \(a\) and \(b\) are known as the parameters of continuous uniform distribution. We cannot…
Discrete Uniform Distribution
A discrete random variable can assume a finite or countable number of values. Put simply, it is possible to list all the outcomes. Remember that a random variable is just a quantity whose future outcomes are not known with certainty….
