Continuous Random Variables

We look at the definitions first.

A continuous random variable, X, has a probability density function (PDF), f(\bullet) if f(x) \ge 0 and for all events A
P(X \in A) = \int_A f(y) dy
The CDF and PDF are related by F(x) = \int_{-\infty}^x f(y) dy
It is good to know that we have P(X \in (x - \frac{\epsilon}{2}, x + \frac{\epsilon}{2})) \approx \epsilon f(x)

We X has a normal distribution, X \sim N(\mu, {\sigma}^2), and f(x) = \frac{1}{\sqrt{2 \pi {\sigma}^2}}e^(-\frac{(x - \mu)^2}{2 {\sigma}^2}). And \mathbb{E}[X] = \mu while Var(X) = {\sigma}^2

Log normal Distribution Source:

Log normal Distribution Source:

We also have the log-normal distribution, X \sim LN (\mu, {\sigma}^2) and log(X) \sim N(\mu, {\sigma}^2). Here, \mathbb{E} [X] = e^(\mu + \frac{{\sigma}^2}{2}) and Var(X) = e^(2 \mu + {\sigma}^2) (e^{{\sigma}^2} - 1). The log-normal distribution is very important in financial applications, for starters, the Black Scholes Equation.

    pingbacks / trackbacks

    Leave a Comment

    Contact Us

    CONTACT US We would love to hear from you. Contact us, or simply hit our personal page for more contact information

    Not readable? Change text. captcha txt

    Start typing and press Enter to search