Question

True or False

(a) If X ∩ Y = ∅ then the two events X and Y are independent?

(b) If event X is independent of event Y, then X^c is independent of Y?

(c) For a discrete random variable X, we have limx->∞ pX(x) = 0?

(d) For a continuous random variable X, we have limx->∞ fX(x) = 0?

(e) For a continuous random variable X, we have limx->0 fX(x) ≤ 1?

(f) For two discrete random variables X and Y , we have ∑x pX,Y (x, y) = 1?

(g) For two discrete random variables X and Y , we have ∑x pX|Y (x|y) = 1?

(h) If random variable A is continuous, then for a real-valued function g, Y = g(A) is a continuous random variable?

(i) If random variable A is discrete, then for a real-valued function g, Y = g(A) is a discrete random variable?

(j) For a random variable A, {A = 0} ∩ {A = 1} = ∅

Answer 1

@ for continuaus rendum venable x, we have Слт х? o }x(+) 219 | Tese pm 1 os fxbus 1 Per two discrete rendom veniable X and Y We have Ex Px,y (24)=1? [false ? Since Es Pxy 18, 9) = 1 X y 9 for two discrete rov X and y ex. Bxly (aly)=1 ? False E(xiy=y), it may take any velle 16 True 0 True 1o True. True

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Sollition o tf Xoy= Then the two events X and qre independent, Fue Y O o interest to chrome se 6 Hleme intersect they are not interseet to each other re Xn you o -> X and Y are indepondent If event y afect is independent o y then fc is independent of y. False.] X and Y are independent Cev/xy=0 su cov( x5, y)= E(*%)-ECXEJElyy to =) ya end y may not independent © for a discrete random vqriable x Where lim Pr (2)=0 24 PX 1920 True for a continualus random vantable X We have om Fla)=0 = for Tove F(a)= P ( xL xj flas de =0 same explanatton for © ,