(1) Let X be a (finite) random variable. Prove that Qx(A) = P(X(A)), A B(R) is...

  

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(1) Let X be a (finite) random variable. Prove that Qx(A) = P(X(A)), A B(R) is a probability measure on B(R). [10 marks] (2) Determine the expectation of the following discrete random variables: (a) Discrete uniformly distributed random variable: px(k) = 1/n, k = 1, 2,..., n. (b) Bernoulli random variable with parameter p: px (0) = p, px(1) 1-p, where 0 (1) Let X be a (finite) random variable. Prove that Qx(A) = P(X(A)), A B(R) is a probability measure on B(R). [10 marks] (2) Determine the expectation of the following discrete random variables: (a) Discrete uniformly distributed random variable: px(k) = 1/n, k = 1, 2,..., n. (b) Bernoulli random variable with parameter p: px (0) = p, px(1) 1-p, where 0