2. (4) Consider the following rule p(X-k) -c()*,k-1;2,..3 for a discrete random

Question

2. (4) Consider the following rule p(X-k) -c()*,k-1;2,..3 for a discrete random variable X. Find the constant c so that the probability rule provides a distribution table. Develop the distribution table. Compute expectation of this random variable. What is the probability that the variable takes a value not less than 3? 2 The answer must include · -the distribution table with values of the random variable in the first row . - respective probabilities according to the formula * - computing c - probabilities in the second row of the table . - computing expectation . - computing the probability of the event

Explanation / Answer

1. The distribution table with values of the random variables

K : 1 2 3

2. Probabilities according to the formula

K : 1 2 3

P(X = k) : c(1/2) c(1/2)^2   c(1/2)^3

3. Since sum of all probabilities are equal to zero

i.e. sum of P(X=k) is 1

i.e. c(1/2) + c(1/2)^2 + c(1/2)^3    =1

c { (1/2) + (1/2)^2 + (1/2)^3 } = 1

c [ 0.5 + 0.25 + 0.125] = 1

c = 1.1429 = 8/7

4  

K : 1 2 3

P(X = k) : 4/7 2/7 1/7

5. E(X) = 1 * 4/7 + 2*2/7 + 3*1/7 = 4/7 + 4/7 + 3/7 = 11/7

6. P(>=3) = P(X=3) = 1/7

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