(Solved): 2. Consider rolling an unfair 4-sided die; that is, each number 1,2,3 and 4 does not have the same ...

2. Consider rolling an unfair 4-sided die; that is, each number 1,2,3 and 4 does not have the same probability of being rolled. Let \( X \) be the number faced up when the die rolls once. The probability mass function, \( f_{X}(x) \) of \( X \) is as follows. \( f_{X}(x)=\left\{\begin{array}{ll}0.2 & \text { if } x=1 \\ 0.1 & \text { if } x=2 \\ 0.3 & \text { if } x=3 \\ 0.4 & \text { if } x=4 \\ 0 & \text { otherwise }\end{array}\right. \) (a) Find the expected value, \( \mathbb{E}[X] \), of \( X \). [2 marks] (b) Find the variance value, \( \operatorname{Var}[X] \), of \( X \). [2 marks] (c) Let \( Y=2 X-1 \). Find \( \operatorname{sd}[Y] \). [2 marks] (d) Write down the cumulative distribution function, \( F_{X}(x) \), of \( X \). [2 marks] (e) Sketch \( F_{X}(x) \) from (d). [2 marks]