Question

Are the random variables Xi and Xj , where i < j, independent? Recall that two random variables X and Y are independent if P[X = k1 ∩Y = k2] = P[X = k1]P[Y = k2] for all k1 and k2. Prove your answer using this definition.

Answer 1

Answer with explanation:

Consider an example of tossing a dice once

Total favorable outcome =6={1,2,3,4,5,6}

  $X_{i}=\text{multiple of 2}={2,4,6}\\\\X_{j}=\text{multiple of 3}={3,6}\\\\P(\text{An Event})=\frac{\text{Total favorable outcome}}{\text{Total favorable outcome}}\\\\P(X_{i}})=\frac{3}{6}\\\\=\frac{1}{2}\\\\P(X_{j}})=\frac{2}{6}\\\\=\frac{1}{3}\\\\\rightarrow X_{i} \cap X_{j}={6}\\\\\rightarrow P(X_{i} \cap X_{j})=\frac{1}{6}\\\\\rightarrow P(X_{i} \cap X_{j})=P(X_{i})\times P(X_{j})$

Hence proved.