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Old 11-20-2017, 02:18 PM
efast efast is offline
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Default expected value

A company agrees to accept the highest of four sealed bids on a property. The four bids are regarded as four independent random variables with common cumulative distribution function
F(x) =
1/2
(1 + sinπx),
3/2 ≤ x ≤ 5/2 and 0 otherwise. What is the expected value of the accepted bid?

My question is how can the four bids be different if they have the same cdf?
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Old 11-20-2017, 02:27 PM
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The same cdf means that (for example) the probability that #1 is less than 2 equals the probability that #2 is less than 2. Compare to flipping 2 fair coins 1 time each and counting the number of heads of each. Each has F(0)=.5, but the results of the actual flips might be different for the two coins.
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Old 11-20-2017, 02:52 PM
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What is the probability that bid #1 is less than 1.6?

What is the probability that bid #2 is at least 1.6 but less than 1.75?

What is the probability that bid #3 is greater than 1.8 but no more than 2.0?

What is the probability that bid #4 is greater than 2.0?
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Old 11-20-2017, 03:38 PM
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Quote:
Originally Posted by efast View Post
A company agrees to accept the highest of four sealed bids on a property. The four bids are regarded as four independent random variables with common cumulative distribution function
F(x) =
1/2
(1 + sinπx),
3/2 ≤ x ≤ 5/2 and 0 otherwise. What is the expected value of the accepted bid?

My question is how can the four bids be different if they have the same cdf?
Reminds me of a story...

I was playing poker with three of my buddies the other night, and we just kept getting dealt the exact same cards! Every time - all four of us!

The other three thought that was strange, but I, being the actuary at the table, calmly explained: it's because we all have the same exact probability distribution, so why wouldn't we all get the exact same cards?
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Old 11-20-2017, 04:45 PM
efast efast is offline
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thank you for the answers.
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Old 11-20-2017, 05:03 PM
Z3ta Z3ta is offline
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Consider the number of heads flips in the flip of a single coin (either 1 or 0).

The PMF is


The number of tails flips in the same flip has this PMF as well.

Both of these random variables have the same PMF (and CDF), but they are necessarily never equal to one another because if the coin lands on heads, then it didn’t land on tails.

My example is two random variables that are completely dependent on one another. In your problem they are specified to be independent. They are independent and identically distributed so they have the same probabilities for values they may realize, but the outcomes of any do not inform the outcomes of the others.
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