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(Dwerete Random Variakle)(25 pts) (a) (5pts) Thue or False No need to justify. Let \( X \) be a discrete random variable with the PMF Px \( (x) \) and CDF \( F(x) \) and finite number of elements in its range. Recall that \( x^{-} \)is the largest value it the tange of \( \mathrm{X} \) that is strictly lese than \( x \). Morcover, if \( x \) is the minimum value in the range, then \( x^{-}=-\infty \) and \( F\left(x^{-}\right)=F(-\infty)=0 \). (i) The CDF of \( \mathrm{X} \) as a continuous function. (Ipt) (ii) \( F(z) \leqslant 1 \quad(t p t) \) (iii) \( P_{x}(z) \leqslant 1 \quad(1 p t) \) (v) \( P[X>a)-1-F(a) \quad(t p t) \) (b) (Spto) Which of the following functions is a \( \mathrm{CDF} \) of a discrete random variable: \[ F_{i}(x)=\left\{\begin{array}{ll} 0 & x<1 \\ 1 / 2 & 13 \end{array} \quad F_{2}(x)=\left\{\begin{array}{ll} 0 & x<0 \\ x^{2} & 0 \leqslant x<1 \\ 1 & x>1 \end{array} \quad F_{3}(x)=\left\{\begin{array}{ll} 0 & x<0 \\ 1 / 2 & 0 \leqslant x<1 \\ 2 / 3 & 1 \leqslant x<2 \\ 5 / 9 & 2 \leqslant x<3 \\ 1 & x \geqslant 3 \end{array}\right.\right.\right. \] Isutrictions Indicated with 'Yos' or 'No'. Juntify your atriswer when it is 'No'. (c) \( (15 \) pes) Suppoen that the PMF of random variable \( X \) is given by \[ P_{X}(x)=\left\{\begin{array}{ll} 1 & x=-2 \\ 1 & x-0 \\ 1 & x-2 \end{array}\right. \] (i) (10pts) Let \( Y=X^{2} \). Find the CDF of random variable \( Y \). (ii) \( (5 p t y) \) Let \( W=(X-2)^{2} \). Find \( E(W) \). Instruction: The oxact number necds to be calculated.

1) The CDF of discrete random variable is continuous. The CDF takes fixed values for a range of input. The CDF of a discrete random variable is, F(x)=