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# Probability Problems including Continous Random Variables

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I have 100 items of a product in stock. The probability mass function for the product's demand D is P(D=90)=P(D=100)=P(D=110)=1/3
a) find the mass function, mean and variance of the number of items sold.
b) find the mass function, mean, and variance of the amount of demand that will be unfilled because of the lack of stock.

##### Solution Summary

The probability problems including continuous random variables are examined.

##### Solution Preview

Let p be the probability that we pick a heart card. Then q=1-p is the probability we pick a non-heart card.
Since we return the card every time to the deck before we draw the next one, the probability to pick a hear each time is p=1/4 and therefore q=1-p=3/4
The probability to have exactly n hearts out of 5 in one configuration is
(1.1)
Since order is not important (we don't care which cards out of the 5 are hearts) The number of different configurations for a certain number of hearts cards in the group is the Binomial coefficient:
(1.2)
Then, the probability to have exactly n heart cards in any group of five cards, regardless of order is:
(1.3)

Thus we can write a little distribution table:
n P(n) Xn Payoff
0
0
1
4
2
8
3
12
4
16
5
20

Note that as expected:
(1.4)

The mean (expected value) of the payoff is:

(1.5)

The variance is:

(1.6)

The density function is:
(1.7)
This function tells us what is the probability that the random variable X attains a certain value between 0 and 4.
Since the random variable must attain some (any) value between 0 and 4, when we add all the probabilities for this to happen we must obtain 1.
In continuous system the summation over all probabilities turns into and integral:
(1.8)

In our case this gives:

(1.9)
And:
...

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