Random Variables (Part 2: Continuous Random Variables)
A continuous random variable is a random variable which can take infinitely many values. The probabilities associated with a continuous RV is defined by probability density function(PDF).
#### Probability Density Function (PDF) :As a continuous RV takes infinite values, the probability \(P(X=x)\) for it can not be defined and takes a value of 0. Instead we define a probability density funaction, which intutively depicts probability per unit space, where space is defined by the range of the underlying random variable. For a continuous RV \(X\) with a PDF \(f(x)\), the probability can be given as:
\[P(a \leq X \leq b) = \int _{a}^{b} f(x) dx\]If we integrate the PDF for the entire range of \(X\), it will be 1, i.e. \(\int _{-\infty}^{\infty} f(x) dx = 1\)
#### Cumulative Distribution Function (Continuous RV) :The cumulative distribution funaction (CDF) of a continuous random variable is given as:
\[F(x) = P(X \leq x) = \int _{-\infty}^{x} f(t) dt\]#### Mean, Median and Variance of Continuous RV :The mean and variance of a continuous RV is defined in a similar way as discrete RV.
\[\mu_X = \int _{-\infty}^{\infty} x f(x) dx\]\[\sigma_X^2 = \int _{-\infty}^{\infty} (x-\mu_X)^2 f(x) dx\]The median is the point which divides the dataset into two equal halves. Hence, it can be calculted by solving the following equation (where \(x_m\) is the median):
\[F(x_m) = P(X \leq x_m) = \int _{-\infty}^{x_m} f(x) dx = 0.5\]#### Linear Functions of Random Variables :If \(X\) is a random variable and \(a\) and \(b\) are constants:
\[\mu _{aX+b} = a\mu_X + b\]\[\sigma _{aX+b}^2 = a^2\sigma_X^2\]\[\sigma _{aX+b} = \lvert a \rvert\sigma_X\]### Reference :https://www.mheducation.com/highered/product/statistics-engineers-scientists-navidi/M0073401331.html