The Cumulative Distribution Function: Interpretation and Determination of Probabilities
A cumulative distribution offers a convenient tool for determining probabilities for a given random variable. As you have already learnt in a previous learning outcome statement, a cumulative distribution function, F(x), gives the probability that the random variable X is…
Discrete Uniform Random Variables, a 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.
Binomial Stock Price Tree
A binomial tree is used to predict stock price movements assuming there are two possible outcomes, each of which has a known probability of occurrence.
Tracking Error
Tracking error refers to the difference in returns between a portfolio (index fund) and a benchmark (target index) against which its performance is evaluated. In other words, it is the difference between the returns on an index fund and the…
Univariate vs. Multivariate Distributions and the Role of Correlation in the Multivariate Normal Distribution
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.
Continuous Uniform Distribution
The continuous uniform distribution is such that the random variable X takes values between α (lower limit) and β (upper limit). In the field of statistics, α and β are known as the parameters of the continuous uniform distribution. We…
Key Properties of the Normal Distribution
A random variable is said to have the 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…
The Standard Normal Distribution: Calculation and Interpretation of Probability
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:
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…
Shortfall Risk, Safety-first Ratio and Selection of an Optimal Portfolio Using Roy’s Safety-first Criterion
Shortfall Risk Shortfall risk refers to the probability that a portfolio will not exceed the minimum (benchmark) return that has been set by an investor. In other words, it is the risk that a portfolio will fall short of the…
