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# (Solved): need help i will thumbs up  (Dwerete Random Variakle)(25 pts) (a) (5pts) Thue or False No ne ...

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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.

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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)=
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