Statistics I20765 marksNumerical: joint probability density functionJoint probability distribution of two random variables
If two random variables have the joint probability density function find (i) constant k (ii) conditional probability density function of X (iii) Identify whether X and Y are independent.
If two random variables have the joint probability density function $$f(x,y) = \begin{cases} k(2x + 3y), & 0 \leq x \leq 1, 0 \leq y \leq 1 \ 0, & \text{otherwise} \end{cases}$$
find (i) constant $k$ (ii) conditional probability density function of $X$ (iii) Identify whether $X$ and $Y$ are independent. [5]
- Joint PDF: $f(x,y) = k(2x+3y)$ for $0 \le x \le 1$, $0 \le y \le 1$; zero otherwise. --- The total probability must equal 1: $$\int0^1 \int0^1 k(2x+3y), dx, dy = 1$$ Inner integral (over x): $$\int0^1 (2x+3y), dx = \left[x^2 + 3xy\right]0^1 = 1 + 3y$$ Outer integral (over y): $$k\int0^1 (1+3...