Question

Which of the following distributions is(are) valid discrete probability distribution(s)? 2. 3. 4. X p(x) X P(X) X p(x) X P(X) 0.3 0 0.3 0 0.2 0 0.1 1 0.4 1 -0.2 1 0.7 1 0.1 2 0.3 2 0.9 2 0.2 N 0.8 O All are valid O 1,3, and 4 only 1 and 4 only 1 only 4 only

Answer 1

solution Flor solving this we need to use arioms of probability. There are 3 anioms for probability. They are are. any ! Aniom 1: For event A., PCA) zo. That is probability is ù always positive Aniom 2 : Probability of the sample space s i p(s) = 1 That is total probability ü equal to one. Aniom 3 ! If A, A2 are disjoint events, then PCA, UAZ V...') = P(A) + P(12) +--, Now let us consider our problem. We have 4 sets of x values and it's pmf. Let us consider individually 1. ) n o 2 P(x-n) 0.310.4 0.3. Here all 2 all P(x) is greater than zelo and EPCX=) - 0.3+0.4+0.3-1. Hence it is a valid discrete probability nd distribution.

2. ) ze 0 2 PX) 0.3 -0.2 0.9. is. Here the P(x) corresponding to X=1 negative. So the first So it is not a valid discrete probability distribution: anion is violated. 3 ) X 12 25 P(X) 10.2 0.7 0.2. Hell all P(x) >0 but the total probability 2. 3 P(x~) Plo) + P (1) +P(2) M-0 0.2 +0.7+0.2 el.lt , P(s) 1, so the second aroom is violated. Hence it i not a probability distributiden 4) Х 0 2 PCX)| 0.1 0. 0.8 Here all p(x) 20 and P(s)= { P(x) = 0.) +0.1 +0.8 = / Hence it is a valid probability distribution so we can say that all required answer 1 and 4 only Coption 3) ين