Question

Based on the Geometric Distribution, given an event with a mean recurrence interval of Tˉ, the probability of not experiencing this event over a series of n events is: P( No ocurrence over Tˉ)=(1−p)Tˉ This is the probability of not ever experiencing this event over it's recurrence interval. Whereas, the probability of experiencing an event within its recurrence interval is: P( Ocurrence over Tˉ)=1−(1−p)Tˉ Perform the following calculations and submit two graphical plots with a written response. 1. For Tˉ=10,20,50,100,1000 years, plot the following relationship: Tˉ vs. P( No ocurrence over Tˉ). Note if and to what value the probability of no occurrence within a time period of Tˉ converges. 2. Do the same for the probability that you have an occurrence within the recurrence interval, Tˉ. This is simply the compliment of the relation you plotted in (1). To do this, you can make a vector for Tˉ and then plot the probability functions given above as a function of Tˉ, where p=1/Tˉ. Note the values these two relations converge on and interpret their meaning using the following example: What is the probability that a home built within the 100 -yr (1% annual exceedance probability, AEP) floodplain will be flooded over 100 years? Hint: this is discussed on page 110-111 of your textbook.

Answer 1

The probability of experiencing an event within its mean recurrence interval Tˉ is given by P(Occurrence over Tˉ) = 1 - (1 - p)^Tˉ, where p = 1/Tˉ. The relationship between Tˉ and these probabilities can be plotted to understand the likelihood of experiencing such an event.

Step-by-step explanation:

The question concerns the Geometric Distribution and its applications in calculating the probability of events such as floods based on their recurrence intervals.

Specifically, students are asked to consider the probability of experiencing an event within its mean recurrence interval (Tˉ) and to plot this relationship for various Tˉ values.

probability of no occurrence is given by P(No occurrence over Tˉ) = (1 - p)^Tˉ, while the occurrence within the interval is P(Occurrence over Tˉ) = 1 - (1 - p)^Tˉ, with p being the reciprocal of Tˉ (p = 1/Tˉ).

Using the example, the probability that a home within a 100-year floodplain will experience a flood within 100 years is the complement of not experiencing a flood.

The probability of not experiencing a flood (P(No flood over 100 years) ) is (1 - 1/100)^100. Therefore, the probability of experiencing at least one flood is P(Flood over 100 years) = 1 - (1 - 1/100)^100.

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