Using the chebychev inequality in the absence of st.dev but known max value
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A random variable X takes the maximum value of 80, and has a mean equal to 50. Give the best upper bound on P(X<=20).
So is it possible to use the Chebychev inequality here. Note that both values are 30 away from the mean. So is it correct to say that because P(X>=80) = 0 , then because P(X<=20) involves a similar calculation (because both values are 30 away from the mean), it would follow that P(X<=20) = 0 as well.
NOTE: We do not have the distribution of the RV.
probability upper-lower-bounds
asked Nov 16 at 19:14
helloworld
477
add a comment |
up vote
0
down vote
favorite
A random variable X takes the maximum value of 80, and has a mean equal to 50. Give the best upper bound on P(X<=20).
So is it possible to use the Chebychev inequality here. Note that both values are 30 away from the mean. So is it correct to say that because P(X>=80) = 0 , then because P(X<=20) involves a similar calculation (because both values are 30 away from the mean), it would follow that P(X<=20) = 0 as well.
NOTE: We do not have the distribution of the RV.
probability upper-lower-bounds
asked Nov 16 at 19:14
helloworld
477
"So is it possible to use the Chebychev inequality here." Erm, no, why would it be?
– Clement C.
Nov 16 at 19:16
(There are r.v.'s satisfying the assumptions which are not even in $L^2$, i.e. do not have a standard deviation. How do you expect to apply Chebyshev's inequality?)
– Clement C.
Nov 16 at 19:20
Either of these three inequalities have to be used: Markov, Chebychev and Chernoff. Since Markov inequality is pretty loose here and the chernoff can only be used for RV's that take the values between 0 and 1, I asked for Chebychev. Do you know another way to get a good upper bound?
– helloworld
Nov 16 at 19:20
You can only use Markov, that's the only one among the three inequalities you list which can apply here given the assumptions.
– Clement C.
Nov 16 at 19:21
Alright, thank you.
– helloworld
Nov 16 at 19:25
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A random variable X takes the maximum value of 80, and has a mean equal to 50. Give the best upper bound on P(X<=20).
So is it possible to use the Chebychev inequality here. Note that both values are 30 away from the mean. So is it correct to say that because P(X>=80) = 0 , then because P(X<=20) involves a similar calculation (because both values are 30 away from the mean), it would follow that P(X<=20) = 0 as well.
NOTE: We do not have the distribution of the RV.
probability upper-lower-bounds
asked Nov 16 at 19:14
helloworld
477
A random variable X takes the maximum value of 80, and has a mean equal to 50. Give the best upper bound on P(X<=20).
So is it possible to use the Chebychev inequality here. Note that both values are 30 away from the mean. So is it correct to say that because P(X>=80) = 0 , then because P(X<=20) involves a similar calculation (because both values are 30 away from the mean), it would follow that P(X<=20) = 0 as well.
NOTE: We do not have the distribution of the RV.
probability upper-lower-bounds
probability upper-lower-bounds
asked Nov 16 at 19:14
helloworld
477
asked Nov 16 at 19:14
helloworld
477
asked Nov 16 at 19:14
helloworld
477
asked Nov 16 at 19:14
helloworld
477
asked Nov 16 at 19:14
helloworld
477
477
"So is it possible to use the Chebychev inequality here." Erm, no, why would it be?
– Clement C.
Nov 16 at 19:16
(There are r.v.'s satisfying the assumptions which are not even in $L^2$, i.e. do not have a standard deviation. How do you expect to apply Chebyshev's inequality?)
– Clement C.
Nov 16 at 19:20
Either of these three inequalities have to be used: Markov, Chebychev and Chernoff. Since Markov inequality is pretty loose here and the chernoff can only be used for RV's that take the values between 0 and 1, I asked for Chebychev. Do you know another way to get a good upper bound?
– helloworld
Nov 16 at 19:20
You can only use Markov, that's the only one among the three inequalities you list which can apply here given the assumptions.
– Clement C.
Nov 16 at 19:21
Alright, thank you.
– helloworld
Nov 16 at 19:25
add a comment |
"So is it possible to use the Chebychev inequality here." Erm, no, why would it be?
– Clement C.
Nov 16 at 19:16
(There are r.v.'s satisfying the assumptions which are not even in $L^2$, i.e. do not have a standard deviation. How do you expect to apply Chebyshev's inequality?)
– Clement C.
Nov 16 at 19:20
Either of these three inequalities have to be used: Markov, Chebychev and Chernoff. Since Markov inequality is pretty loose here and the chernoff can only be used for RV's that take the values between 0 and 1, I asked for Chebychev. Do you know another way to get a good upper bound?
– helloworld
Nov 16 at 19:20
You can only use Markov, that's the only one among the three inequalities you list which can apply here given the assumptions.
– Clement C.
Nov 16 at 19:21
Alright, thank you.
– helloworld
Nov 16 at 19:25
"So is it possible to use the Chebychev inequality here." Erm, no, why would it be?
– Clement C.
Nov 16 at 19:16
"So is it possible to use the Chebychev inequality here." Erm, no, why would it be?
– Clement C.
Nov 16 at 19:16
(There are r.v.'s satisfying the assumptions which are not even in $L^2$, i.e. do not have a standard deviation. How do you expect to apply Chebyshev's inequality?)
– Clement C.
Nov 16 at 19:20
(There are r.v.'s satisfying the assumptions which are not even in $L^2$, i.e. do not have a standard deviation. How do you expect to apply Chebyshev's inequality?)
– Clement C.
Nov 16 at 19:20
Either of these three inequalities have to be used: Markov, Chebychev and Chernoff. Since Markov inequality is pretty loose here and the chernoff can only be used for RV's that take the values between 0 and 1, I asked for Chebychev. Do you know another way to get a good upper bound?
– helloworld
Nov 16 at 19:20
Either of these three inequalities have to be used: Markov, Chebychev and Chernoff. Since Markov inequality is pretty loose here and the chernoff can only be used for RV's that take the values between 0 and 1, I asked for Chebychev. Do you know another way to get a good upper bound?
– helloworld
Nov 16 at 19:20
You can only use Markov, that's the only one among the three inequalities you list which can apply here given the assumptions.
– Clement C.
Nov 16 at 19:21
You can only use Markov, that's the only one among the three inequalities you list which can apply here given the assumptions.
– Clement C.
Nov 16 at 19:21
Alright, thank you.
– helloworld
Nov 16 at 19:25
Alright, thank you.
– helloworld
Nov 16 at 19:25
add a comment |
1 Answer
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You cannot use Chebyshev's inequality here, as you don't even know whether $X$ is square integrable (i.e., $X$ may not even have a standard deviation). (If you want, I can provide an example of such thing.)
So, among the three inequalities you list (Markov, Chebyshev, Chernoff), the only one applicable is the weakest, Markov. Using it on the non-negative r.v. $Y = 80-X$ which has expectation $30$, you get
$$
mathbb{P}{Xleq 20} = mathbb{P}{Ygeq 60} leq frac{mathbb{E}[Y]}{60} = boxed{frac{1}{2}},.
$$
Note further that this bound cannot be improved without further assumptions, as shown by the following example: $X$ taking value $80$ with probability $frac{1}{2}$, and value $20$ with probability $frac{1}{2}$.
answered Nov 16 at 19:28
Clement C.
48.9k33784
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
You cannot use Chebyshev's inequality here, as you don't even know whether $X$ is square integrable (i.e., $X$ may not even have a standard deviation). (If you want, I can provide an example of such thing.)
So, among the three inequalities you list (Markov, Chebyshev, Chernoff), the only one applicable is the weakest, Markov. Using it on the non-negative r.v. $Y = 80-X$ which has expectation $30$, you get
$$
mathbb{P}{Xleq 20} = mathbb{P}{Ygeq 60} leq frac{mathbb{E}[Y]}{60} = boxed{frac{1}{2}},.
$$
Note further that this bound cannot be improved without further assumptions, as shown by the following example: $X$ taking value $80$ with probability $frac{1}{2}$, and value $20$ with probability $frac{1}{2}$.
answered Nov 16 at 19:28
Clement C.
48.9k33784
add a comment |
up vote
1
down vote
accepted
You cannot use Chebyshev's inequality here, as you don't even know whether $X$ is square integrable (i.e., $X$ may not even have a standard deviation). (If you want, I can provide an example of such thing.)
So, among the three inequalities you list (Markov, Chebyshev, Chernoff), the only one applicable is the weakest, Markov. Using it on the non-negative r.v. $Y = 80-X$ which has expectation $30$, you get
$$
mathbb{P}{Xleq 20} = mathbb{P}{Ygeq 60} leq frac{mathbb{E}[Y]}{60} = boxed{frac{1}{2}},.
$$
Note further that this bound cannot be improved without further assumptions, as shown by the following example: $X$ taking value $80$ with probability $frac{1}{2}$, and value $20$ with probability $frac{1}{2}$.
answered Nov 16 at 19:28
Clement C.
48.9k33784
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
You cannot use Chebyshev's inequality here, as you don't even know whether $X$ is square integrable (i.e., $X$ may not even have a standard deviation). (If you want, I can provide an example of such thing.)
So, among the three inequalities you list (Markov, Chebyshev, Chernoff), the only one applicable is the weakest, Markov. Using it on the non-negative r.v. $Y = 80-X$ which has expectation $30$, you get
$$
mathbb{P}{Xleq 20} = mathbb{P}{Ygeq 60} leq frac{mathbb{E}[Y]}{60} = boxed{frac{1}{2}},.
$$
Note further that this bound cannot be improved without further assumptions, as shown by the following example: $X$ taking value $80$ with probability $frac{1}{2}$, and value $20$ with probability $frac{1}{2}$.
answered Nov 16 at 19:28
Clement C.
48.9k33784
You cannot use Chebyshev's inequality here, as you don't even know whether $X$ is square integrable (i.e., $X$ may not even have a standard deviation). (If you want, I can provide an example of such thing.)
So, among the three inequalities you list (Markov, Chebyshev, Chernoff), the only one applicable is the weakest, Markov. Using it on the non-negative r.v. $Y = 80-X$ which has expectation $30$, you get
$$
mathbb{P}{Xleq 20} = mathbb{P}{Ygeq 60} leq frac{mathbb{E}[Y]}{60} = boxed{frac{1}{2}},.
$$
Note further that this bound cannot be improved without further assumptions, as shown by the following example: $X$ taking value $80$ with probability $frac{1}{2}$, and value $20$ with probability $frac{1}{2}$.
answered Nov 16 at 19:28
Clement C.
48.9k33784
answered Nov 16 at 19:28
Clement C.
48.9k33784
answered Nov 16 at 19:28
Clement C.
48.9k33784
answered Nov 16 at 19:28
Clement C.
48.9k33784
48.9k33784
add a comment |
add a comment |
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"So is it possible to use the Chebychev inequality here." Erm, no, why would it be?
– Clement C.
Nov 16 at 19:16
(There are r.v.'s satisfying the assumptions which are not even in $L^2$, i.e. do not have a standard deviation. How do you expect to apply Chebyshev's inequality?)
– Clement C.
Nov 16 at 19:20
Either of these three inequalities have to be used: Markov, Chebychev and Chernoff. Since Markov inequality is pretty loose here and the chernoff can only be used for RV's that take the values between 0 and 1, I asked for Chebychev. Do you know another way to get a good upper bound?
– helloworld
Nov 16 at 19:20
You can only use Markov, that's the only one among the three inequalities you list which can apply here given the assumptions.
– Clement C.
Nov 16 at 19:21
Alright, thank you.
– helloworld
Nov 16 at 19:25