Exponential Distruproblem
Four tigers in a reserve forest are monitored using geo tags. The waiting times for responses from 4 tigers in the reserve follow an iid exponential distribution with mean 3. If the system has to locate all 4 tigers within 5 minutes, it has to reduce the expected response time of each geo tag. What is the maximum expected response time that will produce a location for all four tigers within 5 minutes or less with at least $90%$ probability.
I don’t understand this question, can someone explain what it is asking for and provide the solution?
probability probability-distributions random-variables exponential-distribution
edited Dec 4 '18 at 9:40
Tki Deneb
29210
asked Dec 3 '18 at 20:48
user601297
1226
add a comment |
Four tigers in a reserve forest are monitored using geo tags. The waiting times for responses from 4 tigers in the reserve follow an iid exponential distribution with mean 3. If the system has to locate all 4 tigers within 5 minutes, it has to reduce the expected response time of each geo tag. What is the maximum expected response time that will produce a location for all four tigers within 5 minutes or less with at least $90%$ probability.
I don’t understand this question, can someone explain what it is asking for and provide the solution?
probability probability-distributions random-variables exponential-distribution
edited Dec 4 '18 at 9:40
Tki Deneb
29210
asked Dec 3 '18 at 20:48
user601297
1226
Let $X_1,dots,X_4$ be the response time for the $4$ tigers. Those random variables are iid exponentially distributed with mean $3$. Now we have $$ P(max(X_1,dots,X_4) leq 5) < 0.9. $$ But you want that probability to be $geq 0.9$. To achive that, you can change the mean of the exponential distributions to any $mu in mathbb{R}$. How big can you choose $mu$ such that the above probability is $geq 0.9$?
– Tki Deneb
Dec 4 '18 at 9:37
add a comment |
Four tigers in a reserve forest are monitored using geo tags. The waiting times for responses from 4 tigers in the reserve follow an iid exponential distribution with mean 3. If the system has to locate all 4 tigers within 5 minutes, it has to reduce the expected response time of each geo tag. What is the maximum expected response time that will produce a location for all four tigers within 5 minutes or less with at least $90%$ probability.
I don’t understand this question, can someone explain what it is asking for and provide the solution?
probability probability-distributions random-variables exponential-distribution
edited Dec 4 '18 at 9:40
Tki Deneb
29210
asked Dec 3 '18 at 20:48
user601297
1226
Four tigers in a reserve forest are monitored using geo tags. The waiting times for responses from 4 tigers in the reserve follow an iid exponential distribution with mean 3. If the system has to locate all 4 tigers within 5 minutes, it has to reduce the expected response time of each geo tag. What is the maximum expected response time that will produce a location for all four tigers within 5 minutes or less with at least $90%$ probability.
I don’t understand this question, can someone explain what it is asking for and provide the solution?
probability probability-distributions random-variables exponential-distribution
probability probability-distributions random-variables exponential-distribution
edited Dec 4 '18 at 9:40
Tki Deneb
29210
asked Dec 3 '18 at 20:48
user601297
1226
edited Dec 4 '18 at 9:40
Tki Deneb
29210
asked Dec 3 '18 at 20:48
user601297
1226
edited Dec 4 '18 at 9:40
Tki Deneb
29210
edited Dec 4 '18 at 9:40
Tki Deneb
29210
edited Dec 4 '18 at 9:40
Tki Deneb
29210
29210
asked Dec 3 '18 at 20:48
user601297
1226
asked Dec 3 '18 at 20:48
user601297
1226
asked Dec 3 '18 at 20:48
user601297
1226
1226
Let $X_1,dots,X_4$ be the response time for the $4$ tigers. Those random variables are iid exponentially distributed with mean $3$. Now we have $$ P(max(X_1,dots,X_4) leq 5) < 0.9. $$ But you want that probability to be $geq 0.9$. To achive that, you can change the mean of the exponential distributions to any $mu in mathbb{R}$. How big can you choose $mu$ such that the above probability is $geq 0.9$?
– Tki Deneb
Dec 4 '18 at 9:37
add a comment |
Let $X_1,dots,X_4$ be the response time for the $4$ tigers. Those random variables are iid exponentially distributed with mean $3$. Now we have $$ P(max(X_1,dots,X_4) leq 5) < 0.9. $$ But you want that probability to be $geq 0.9$. To achive that, you can change the mean of the exponential distributions to any $mu in mathbb{R}$. How big can you choose $mu$ such that the above probability is $geq 0.9$?
– Tki Deneb
Dec 4 '18 at 9:37
Let $X_1,dots,X_4$ be the response time for the $4$ tigers. Those random variables are iid exponentially distributed with mean $3$. Now we have $$ P(max(X_1,dots,X_4) leq 5) < 0.9. $$ But you want that probability to be $geq 0.9$. To achive that, you can change the mean of the exponential distributions to any $mu in mathbb{R}$. How big can you choose $mu$ such that the above probability is $geq 0.9$?
– Tki Deneb
Dec 4 '18 at 9:37
Let $X_1,dots,X_4$ be the response time for the $4$ tigers. Those random variables are iid exponentially distributed with mean $3$. Now we have $$ P(max(X_1,dots,X_4) leq 5) < 0.9. $$ But you want that probability to be $geq 0.9$. To achive that, you can change the mean of the exponential distributions to any $mu in mathbb{R}$. How big can you choose $mu$ such that the above probability is $geq 0.9$?
– Tki Deneb
Dec 4 '18 at 9:37
add a comment |
1 Answer
1
active
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votes
The first step is to find the distribution of the max of exponential RVs. Let $X_n sim exp(lambda)$ be i.i.d. RVs and let $Y = max_n {X_n}$.
$P(Y leq y)$
$=P(max_n {X_n} leq y)$
$=P(X_1 leq y, ; X_2 leq y, ; ..., ; X_n leq y)$
$=P(X_1 leq y)^n$ (using independence)
$=(1-e^{-lambda y})^n$ (using definition of exponential CDF)
In your example, $n=4$, so you can compute $lambda$ from the inequality $P(Y leq 5) geq 0.9$, i.e. $(1-e^{-5 lambda})^4 geq 0.9$.
answered Dec 6 '18 at 1:33
Aditya Dua
83918
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
The first step is to find the distribution of the max of exponential RVs. Let $X_n sim exp(lambda)$ be i.i.d. RVs and let $Y = max_n {X_n}$.
$P(Y leq y)$
$=P(max_n {X_n} leq y)$
$=P(X_1 leq y, ; X_2 leq y, ; ..., ; X_n leq y)$
$=P(X_1 leq y)^n$ (using independence)
$=(1-e^{-lambda y})^n$ (using definition of exponential CDF)
In your example, $n=4$, so you can compute $lambda$ from the inequality $P(Y leq 5) geq 0.9$, i.e. $(1-e^{-5 lambda})^4 geq 0.9$.
answered Dec 6 '18 at 1:33
Aditya Dua
83918
add a comment |
The first step is to find the distribution of the max of exponential RVs. Let $X_n sim exp(lambda)$ be i.i.d. RVs and let $Y = max_n {X_n}$.
$P(Y leq y)$
$=P(max_n {X_n} leq y)$
$=P(X_1 leq y, ; X_2 leq y, ; ..., ; X_n leq y)$
$=P(X_1 leq y)^n$ (using independence)
$=(1-e^{-lambda y})^n$ (using definition of exponential CDF)
In your example, $n=4$, so you can compute $lambda$ from the inequality $P(Y leq 5) geq 0.9$, i.e. $(1-e^{-5 lambda})^4 geq 0.9$.
answered Dec 6 '18 at 1:33
Aditya Dua
83918
add a comment |
The first step is to find the distribution of the max of exponential RVs. Let $X_n sim exp(lambda)$ be i.i.d. RVs and let $Y = max_n {X_n}$.
$P(Y leq y)$
$=P(max_n {X_n} leq y)$
$=P(X_1 leq y, ; X_2 leq y, ; ..., ; X_n leq y)$
$=P(X_1 leq y)^n$ (using independence)
$=(1-e^{-lambda y})^n$ (using definition of exponential CDF)
In your example, $n=4$, so you can compute $lambda$ from the inequality $P(Y leq 5) geq 0.9$, i.e. $(1-e^{-5 lambda})^4 geq 0.9$.
answered Dec 6 '18 at 1:33
Aditya Dua
83918
The first step is to find the distribution of the max of exponential RVs. Let $X_n sim exp(lambda)$ be i.i.d. RVs and let $Y = max_n {X_n}$.
$P(Y leq y)$
$=P(max_n {X_n} leq y)$
$=P(X_1 leq y, ; X_2 leq y, ; ..., ; X_n leq y)$
$=P(X_1 leq y)^n$ (using independence)
$=(1-e^{-lambda y})^n$ (using definition of exponential CDF)
In your example, $n=4$, so you can compute $lambda$ from the inequality $P(Y leq 5) geq 0.9$, i.e. $(1-e^{-5 lambda})^4 geq 0.9$.
answered Dec 6 '18 at 1:33
Aditya Dua
83918
answered Dec 6 '18 at 1:33
Aditya Dua
83918
answered Dec 6 '18 at 1:33
Aditya Dua
83918
answered Dec 6 '18 at 1:33
Aditya Dua
83918
83918
add a comment |
add a comment |
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Let $X_1,dots,X_4$ be the response time for the $4$ tigers. Those random variables are iid exponentially distributed with mean $3$. Now we have $$ P(max(X_1,dots,X_4) leq 5) < 0.9. $$ But you want that probability to be $geq 0.9$. To achive that, you can change the mean of the exponential distributions to any $mu in mathbb{R}$. How big can you choose $mu$ such that the above probability is $geq 0.9$?
– Tki Deneb
Dec 4 '18 at 9:37