Consider an experiment having three possible outcomes Outcome 1, Outcome 2 and Outcome 3, so that...

  

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Consider an experiment having three possible outcomes Outcome 1, Outcome 2 and Outcome 3, so that Outcome 1 occurs with probability pi, Outcome 2 occurs with probability Pz and Outcome 3 occurs with probability p3. Suppose independent repetitions of the experiment are made. (a) Let the random variable T denote the number of trials until Outcome 3 occurs. State the distribution of T. [2 marks] (b) Let the random variable X denote the number of occurences of Outcome 1 before Outcome 3 occurs. Compute the probability mass function of X. [4 marks] (c) Compute the probability that Outcome 1 occurs before Outcome 3. [2 marks] (d) Let the random variable Y denote the number of trials until both Outcome 1 and Outcome 3 occur for the first time. State the probability mass function of Y. Justify your answer. [4 marks] You may use that j! Em(m – 1)(m – 2)... (m- (j- 1)) am-i = lal < 1,j €N. (1- a)+1 1. lal < 1. m=0 Consider an experiment having three possible outcomes Outcome 1, Outcome 2 and Outcome 3, so that Outcome 1 occurs with probability pi, Outcome 2 occurs with probability Pz and Outcome 3 occurs with probability p3. Suppose independent repetitions of the experiment are made. (a) Let the random variable T denote the number of trials until Outcome 3 occurs. State the distribution of T. [2 marks] (b) Let the random variable X denote the number of occurences of Outcome 1 before Outcome 3 occurs. Compute the probability mass function of X. [4 marks] (c) Compute the probability that Outcome 1 occurs before Outcome 3. [2 marks] (d) Let the random variable Y denote the number of trials until both Outcome 1 and Outcome 3 occur for the first time. State the probability mass function of Y. Justify your answer. [4 marks] You may use that j! Em(m – 1)(m – 2)... (m- (j- 1)) am-i = lal < 1,j €N. (1- a)+1 1. lal < 1. m=0