(Solved): Let \( X \) and \( Y \) be continuous random variables with joint probability density function giv ...

Let \( X \) and \( Y \) be continuous random variables with joint probability density function given by \[ f(x)=\left\{\begin{array}{ll} 21 x^{2} y^{6} & \text { if } 0 \leq x \leq 1,0 \leq y \leq 1 \\ 0 & \text { otherwise } \end{array}\right. \] Find the following: (a) The probability that \( X \) is less than \( Y \) : \[ P(X1 / 2)= \] (c) The expected values of \( X \) and \( X^{2} \) : \[ \begin{array}{l} E(X)= \\ E\left(X^{2}\right)= \end{array} \] (d) The variance and standard deviation of \( X \) : \[ \begin{array}{l} \operatorname{var}(X)= \\ \sigma(X)= \end{array} \]