Chapter 2 Section 3 Question 6

Question

Let $Z$ ~ $Geometric(\theta )$. Compute $P(5 \leq Z \leq 9)$.

Answer

Author: Mohammad-Ali Bandzar| Date:Oct 15 2020

From Example 2.3.4 in the book we know: $$p\scriptsize{Z}\normalsize{(z)=(1-\theta )^z\theta}$$ Now all we must do is add up the geometic probabilities for Z=5,6,7,8,9 together $$P(5 \leq Z \leq 9)=(1-\theta )^5\theta+(1-\theta )^6\theta+(1-\theta )^7\theta+(1-\theta )^8\theta+(1-\theta )^9\theta$$ That is the probability that Z will be between 5 and 9 inclusively, please note how Z can only be a positive integer as it represents the number of successful trials before a failure

From Example 2.3.4 in the book we know: $$p\scriptsize{Z}\normalsize{(z)=(1-\theta )^z\theta}$$ Now all we must do is add up the geometic probabilities for Z=5,6,7,8,9 together $$P(5 \leq Z \leq 9)=(1-\theta )^5\theta+(1-\theta )^6\theta+(1-\theta )^7\theta+(1-\theta )^8\theta+(1-\theta )^9\theta$$ That is the probability that Z will be between 5 and 9 inclusively, please note how Z can only be a positive integer as it represents the number of successful trials before a failure