Solutions for exercises Download
Solutions for end sem exam Download
In the lecture notes, I have not mentioned for independent random variables $X$, $Y$ for joint distributions. Please follow from here.
Two discrete random variables $X$ and $Y$ are independent if $$ P_{X Y}(x, y)=P_X(x) P_Y(y), \quad \text { for all } x, y $$ Equivalently, $X$ and $Y$ are independent if $$ F_{X Y}(x, y)=F_X(x) F_Y(y), \quad \text { for all } x, y $$
Two continuous random variables $X$ and $Y$ are independent if $$ f_{X Y}(x, y)=f_X(x) f_Y(y), \quad \text { for all } x, y $$ Equivalently, $X$ and $Y$ are independent if $$ F_{X Y}(x, y)=F_X(x) F_Y(y), \quad \text { for all } x, y $$
If $X$ and $Y$ are independent, we have $$ \begin{aligned} & E[X Y]=E[X]E[Y], \\ & E[g(X) h(Y)]=E[g(X)]E[h(Y)] \end{aligned} $$