# Thread: Bounded in Probability proof

1. ## Bounded in Probability proof

Hi all,

I could use some help understanding the following proof so please feel free to pitch in!

Consider any random variable with distribution function . Then given we can bound in the following way. Because the lower limit of is 0 and its upper limit is 1, we can find such that:

for and for

Let then

What I need help understanding is where the inequality in the last line comes from precisely. I know it has something to do with the bounds we set up above but I cannot figure it. I am a beginner so please explain in detail. Thanks.

2. ## Re: Bounded in Probability proof

Note that
1a) , so
Thus,
2a) , or equivalently

On the other hand,
1b)
Thus,
2b)

Add (2a) and (2b)

3. ## The Following User Says Thank You to Mean Joe For This Useful Post:

JohnK (10-02-2013)

4. ## Re: Bounded in Probability proof

Originally Posted by Mean Joe
Note that
1a) , so
Thus,
2a) , or equivalently

On the other hand,
1b)
Thus,
2b)

Add (2a) and (2b)
Wow, thanks a lot. One thing though, in 1a) don't you mean ? That is also implied by the inequality and I think this is the result we need.

5. ## Re: Bounded in Probability proof

I think it should be like

6. ## The Following User Says Thank You to BGM For This Useful Post:

JohnK (10-02-2013)

7. ## Re: Bounded in Probability proof

Originally Posted by BGM
I think it should be like

Why the absolute value in ?

8. ## Re: Bounded in Probability proof

It is because

9. ## The Following User Says Thank You to BGM For This Useful Post:

JohnK (10-02-2013)

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