A person A is supposed to send information to a person B. The information consis

Question

A person A is supposed to send information to a person B. The information consists of two digit numbers, including possibilities where the first digit is 0. For example 07. A is allowed to send only one two digit number to the other side. Person B is then asked to recover the two original numbers just from the received number knowledge of the algorithm person A was using to calculate that number. Person A calculates the average of the two numbers and then send it two the person B. If the average turns not to be an integer then A rounds it upward. For example if the two numbers are 20 and 29, the average is 24.5 and A would send 25 to B. Now assume that every pair of numbers is equally likely, that is it comes with the probability of 1 /10000. We define a random variable X to be the number of possibilities B has to choose from to guess the two original numbers. For example if B receives 03 then there are 13 possibilities, (00,06),(01,05),(02,04),(03,03),(04,02),(05,01),(06,00),(00,05),(01,04),(02,03),(03,02),(04 ,01),(05,00). The problem is to compute the expectation of X.

Explanation / Answer

X be the number of posibilities

E(X) = x1P1 + x2P2 + x3P3 + . . . + xnPn

so for average to be 01.

The numbers can be (01,01)(00,02)(02,00) so the probability will be 1/3

average to be 02 the numbers can be (00,04)(04,00)(01,03)(03,01)(02,02), the probabilty will be 1/5

the average to be 03 the numbers can be as given in the question, the probability will be 1/13

similarly we can find for all the numbers and can find the expectation.

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