Chapter 32. Exponential Random Variables b. What does X represent in this scenar
ID: 3065200 • Letter: C
Question
Chapter 32. Exponential Random Variables b. What does X represent in this scenario? c. What is the parameter? d. What is the expected length of time (in years) between now and when the next hurricane that is category 4 or stronger? e. What is the variance in this length of time? f. What is the probability density function for the length of time before the next hurricane that is category 4 or stronger? Write your answer in function form and show a graph. category 4 or stronger? Write your answer in function form and show a graph category 4 or stronger, during the next 3 years? or stronger, during the period that is between 5 to 10 years from now? the next 3 years, what is the probability that there will not be any during the g. What is the CDF for the length of time before the next hurricane that is h. What is the probability that there will not be any hurricane that is i. W is the probability that there will be a hurricane that is category 4 hat j. Given that there are no hurricanes that are category 4 or stronger during next 10 years? k. How long a waiting time do we need, if we want to be 75% sure that is a hurricane that is category 4 or stronger during the waiting time?Explanation / Answer
QUestion 4 (a) In the study of continuous-time stochastic processes, the exponential distribution is usually used to model the time until something happens in the process. So, here we are waiting for something to happen so we will use exponential process here.
(b) Here X represent the time (in years) before the next hurricane comes that is category 4 or stronger.
(c) Here Parameter for exponential distribution is = 1/6 years-1
(d) Expected length of time between now and in between next hurricane occur = 6 year
(e) Variance of length of time between now and in between next hurricane occur = 6 year2
(f) HEre pdf of x is
f(X) = (1/6) e-x/6 x> 0
(g) Here CDF of x is
F(x) = 1 - e-x/6 ; x > 0
(h) there is no hurricane occur in the three years.
F(x > 3) = 1 - (1 - e-3/6) = 0.6065
(i) So in between 5 to 10 years, there are 5 years so the probability that a hurricane occur in between 5 to 10 years.
Pr(one hurricane occur in between 5 to 10 years) = 1 - Pr(no hurricance occur in between 5 to 10 years)
Pr(no hurricance occur in between 5 to 10 years) = Pr(x > 10 year l x > 5 year) = [ {1- (1 - e-10/6)}/ {1 -(1 - e-5/6)}]
= e-5/6 = 0.4346
Pr(one hurricane occur in between 5 to 10 years) = 1 - 0.4346 = 0.5654
(j) Pr(No hurricane in next 10 years l no hurricane in next 3 years) = Pr(x > 10 years l x > 3 years) = e-10/6/e-3/6 = e-7/6 = 0.3114
(h) Here we have to find the value of X for which F(x) > 0.75
so,
F(x) = 1 - e-x/6
1 - e-x/6 > 0.75
e-x/6 < 0.25
x > 8.32 years
So in the period of 8.32 years we can be 75% sure that there will a hurricane.
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