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6. Determine whether each of the following is true or false (note: the statement is true...

6. Determine whether each of the following is true or false (note: the statement is true if it is always true, otherwise it is false). If you say it is true then refer to a known result or give a proof, while if you say it is false then give a counterexample, i.e., a particular case where it fails. (a) If A, B and C are independent, the Pr(AlBnc)- Pr (A) (b) The events S., A are independent (S is the certain event, ф s the impossible event and A is an arbitrary event here) in (c) If the events A and B are mutually exclusive and Pr (A), Pr (B) are both positive, then they cannot be independent 1 for 0 kKn, n2 1 (f) Let X be a continuous randon variable with pdf fx (x). Then Pr (X = zo) = fx (ro) (g) Let X be a continuous random variable with pdf fx (r) . Then (for a b) Pr (a X b) = x (x)dr (h) Let X be a discrete random variable with pmf px (xi). Then Pr (X-opx (ro) (i) Let X be a discrete random variable with pmf px ().Then (for a S b) Pr (a SXSb) G) Let X be a random variable with edf Fx (x).Then (for a S b) Pr (a SXS b)-Fx (b)- Fx (a)

6. Determine whether each of the following is true or false (note: the statement is true if it is always true, otherwise it is false). If you say it is true then refer to a known result or give a proof, while if you say it is false then give a counterexample, i.e., a particular case where it fails. (a) If A, B and C are independent, the Pr(AlBnc)- Pr (A) (b) The events S., A are independent (S is the certain event, ф s the impossible event and A is an arbitrary event here) in (c) If the events A and B are mutually exclusive and Pr (A), Pr (B) are both positive, then they cannot be independent 1 for 0 kKn, n2 1 (f) Let X be a continuous randon variable with pdf fx (x). Then Pr (X = zo) = fx (ro) (g) Let X be a continuous random variable with pdf fx (r) . Then (for a b) Pr (a X b) = x (x)dr (h) Let X be a discrete random variable with pmf px (xi). Then Pr (X-opx (ro) (i) Let X be a discrete random variable with pmf px ().Then (for a S b) Pr (a SXSb) G) Let X be a random variable with edf Fx (x).Then (for a S b) Pr (a SXS b)-Fx (b)- Fx (a)
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