Thread: Using Weak law of Large numbers

1. Using Weak law of Large numbers

Hello,
To give a brief background
in my college math course on probability and statistics, the professor just finished going over Markov's inequality, Chebychev's inequality, and the Weak law of large numbers.

I tried to figure this problem out for hours, but couldn't understand it. It is for a homework that has already been turned in (ie, this is not for a grade), but I desperately want to know the answer + explanation.

the problem is in the attachment

Any help will be much appreciated!

2. Originally Posted by johnj77
Hello,
To give a brief background
in my college math course on probability and statistics, the professor just finished going over Markov's inequality, Chebychev's inequality, and the Weak law of large numbers.

I tried to figure this problem out for hours, but couldn't understand it. It is for a homework that has already been turned in (ie, this is not for a grade), but I desperately want to know the answer + explanation.

the problem is in the attachment

Any help will be much appreciated!
if i remember it well, $\frac{1}{n} \sum_{i=1}^n X_i$ as $n \rightarrow \infty$ estimates the area of the function $Y_i = g(X_i) \, \forall i=1,...,n$

3. Originally Posted by johnj77
Hello,
To give a brief background
in my college math course on probability and statistics, the professor just finished going over Markov's inequality, Chebychev's inequality, and the Weak law of large numbers.

I tried to figure this problem out for hours, but couldn't understand it. It is for a homework that has already been turned in (ie, this is not for a grade), but I desperately want to know the answer + explanation.

the problem is in the attachment

Any help will be much appreciated!
The $X_i$'s are independent identicaly distributed so:

$\lim_{N \to \infty} \frac{1}{N}\sum_{i=1}^NX_i = \bar{X}$

with probability 1.

Also $\bar{X}=1/2$

RonL

4. I'm sorry, my math in this area is still weak. Captain Black, could you offer a more detailed
explanation? In the problem what does P(Xi > x)= e^(-2x) have to do with anything? how did you get

$
\bar{X}=1/2
$
?

(also, sorry about double posting earlier, I shouldn't have done that)

5. Originally Posted by johnj77
I'm sorry, my math in this area is still weak. Captain Black, could you offer a more detailed
explanation? In the problem what does P(Xi > x)= e^(-2x) have to do with anything? how did you get
It is the complementary cumulative distribution of the X's. It is used to find
the pdf for the X's which is used to find the mean:

$
\bar{X}=1/2
$
?

(also, sorry about double posting earlier, I shouldn't have done that)
RonL