(Solved): a) The distribution (CDF) of a discrete random variable is shown in the figure below. i) Fi ...

a) The distribution (CDF) of a discrete random variable is shown in the figure below. i) Find the probability \\( \\operatorname{Pr}(X \\leq 2) \\). ii) Calculate the variance of \\( X \\), ie VAR \\( [X] \\). iii) Sketch the corresponding probability mass function PMF.\r\nb) The PDF of a random variable \\( X \\) is given by: \\[ f_{X}(x)=\\left\\{\\begin{array}{cc} 0.4+k x, & 0 \\leq x \\leq 4 \\\\ 0 & \\text { otherwise } \\end{array}\\right. \\] i) Find the value \\( k \\) that makes \\( f_{x} \\) a valid PDF. ii) Find \\( P(X>1 / 2) \\). iii) Find the CDF, Fx(x).