9 For the following discrete random variable X with probabil

9. For the following discrete random variable X with probability distribution:

   X     0 1 2 3

P(X) 0.2 0.4 0.15 0.25

(a) Find the probability distribution for Y = 3X^2-2X + 1

(i.e, all values of Y and P(Y )).

(b) Find E[Y ] using part (a).

(c) Find E[X] and E[X2].

(d) Find r(Y less than or equal 2).

(e) Using part (c), verify your result from part (b) using E[X] and E[X^2]

Solution

Y = 3x^2-2x+1

X=0,1,2,3

Y(X) = 1, 2, 9, 22

P(Y) 0.2, 0.4, 0.15, 0.25

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b) E(Y) = sum of yp(y) = 0.2+0.8+1.35+5.5

= 7,85

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c) E(X) = sum of xp(x) = 0.4+0.30+0.75 = 1.45

E(X²) = 0.4+0.6+2.25 = 3.25

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d) P(Y<=2) = 0.2+0.4 =0.6

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e) E(Y) = 3E(x²)-2E(X)+1 = 9.75-2.90+1

= 7.85

This is the same as arrived from pdf of Y.