Hi all,
I have a question that I can't find answer to:
I have 10 random variables X1, X2....X10 which are all independent and exponentially distributed with parameter=2
Xi~exp(2) for i between 1 and 10.
Now the argument says that the probability that the sum of all 10 X's is larger than 10 is less than 0.1.
P(sum of all X's > 10)<=0.1
if I calculate it with direct calculation I am doing it that way:
P(sum of all X's > 10) = P(X1+X2+...+X10 > 10) = P(X1+X1+...+X1 > 10) =
P(10X1 > 10) = P(X1 > 1) = 1 - P(X1 <= 1) =
1 - [integral from 0 to 1 of (2e^(-2*t)dt)]=
1 - 2((1/-2)*e^(-2*t)from 0 to 1) = 1 + (e^(-2) - e^(0)) = 1 + e^(-2) -1 =
e^(-2) = 0.13
and we get the answer that the argument is wrong( 0.1 < 0.13 ).
But, if we calculate the probability using Chebeyshev's inequality:
P(|X1+X2+...+X10 - E(X1+...+X10)| => 10 - E(X1+...+X10)) < V(X1+...+X10)/[10 - E(X1+...+X10)]^2
calculating E(X1+...+X10) = 10/2 = 5
calculating V(X1+...+X10) = 10/2^2 = 10/4 = 2.5
and now we can calculate
P(|X1+X2+...+X10 - E(X1+...+X10)| => 10 - E(X1+...+X10)) < 2.5/25 = 0.1
and we get that the argument is correct (0.1 <= 0.1)
I would really appreciate if someone could tell me where I am wrong.
Thanks !
I have a question that I can't find answer to:
I have 10 random variables X1, X2....X10 which are all independent and exponentially distributed with parameter=2
Xi~exp(2) for i between 1 and 10.
Now the argument says that the probability that the sum of all 10 X's is larger than 10 is less than 0.1.
P(sum of all X's > 10)<=0.1
if I calculate it with direct calculation I am doing it that way:
P(sum of all X's > 10) = P(X1+X2+...+X10 > 10) = P(X1+X1+...+X1 > 10) =
P(10X1 > 10) = P(X1 > 1) = 1 - P(X1 <= 1) =
1 - [integral from 0 to 1 of (2e^(-2*t)dt)]=
1 - 2((1/-2)*e^(-2*t)from 0 to 1) = 1 + (e^(-2) - e^(0)) = 1 + e^(-2) -1 =
e^(-2) = 0.13
and we get the answer that the argument is wrong( 0.1 < 0.13 ).
But, if we calculate the probability using Chebeyshev's inequality:
P(|X1+X2+...+X10 - E(X1+...+X10)| => 10 - E(X1+...+X10)) < V(X1+...+X10)/[10 - E(X1+...+X10)]^2
calculating E(X1+...+X10) = 10/2 = 5
calculating V(X1+...+X10) = 10/2^2 = 10/4 = 2.5
and now we can calculate
P(|X1+X2+...+X10 - E(X1+...+X10)| => 10 - E(X1+...+X10)) < 2.5/25 = 0.1
and we get that the argument is correct (0.1 <= 0.1)
I would really appreciate if someone could tell me where I am wrong.
Thanks !