961 Words4 Pages

Birthday Paradox:
The problem is to find that how many people must be there in a room for having same birthday.
Than what is paradox in it? Paradox is that you may think that the possibility will only come when there are many (lets say 365) people in the room. But you will see that the probability will occur even for few people.
To solve this problem we index the total number of the people in the room as
1,2,3….k where k is equal to the total number of people in room.
And also let us assume that all years has 365 days (ignoring the issue of leap years). Lets say n=365 days
For
i=1,2,3,…,k let bi be the day on which the birthday of ith person occurs, assuming 1< bi *…show more content…*

Analysis Using Indicator Variable: Now we will use indicator variable to calculate an approximately simpler solution to this problem i.e. birthday paradox. We define, for each pair of people (i , j) in the room out of total people k, let us define an indicator variable: Xij indicator random variable, for 1< i < j < k so, Xij = I (person i and j having the same birthday) = Now applying expectancy on both side of the above equation, we have E(Xij) = Pr (person i and j having the same birthday) = 1/n (as calculated previously) So, for the total, say X be the random variable that counts total number of pairs having same birthday. X = ∑_(i=1)^k ∑_(j=i+1)^k Xij Taking expectations on both sides, we get E (X )=E ( ∑_(i=1)^k ∑_(j=i+1)^k Xij ) = ( ∑_(i=1)^k ∑_(j=i+1)^k E (Xij) Solving the summation mathematically we get E (X )= k(k-1)/2n If k(k-1) > 2n than the number of expected pairs having birthday on the same day will may be greater or equal to one. Solving k(k-1) > 2n we get: r = √2n + 1 Now, for example if we have n=365 and k=28 we get by using equation k(k-1)/2n => 28(27)/2(365) =1.0368 Thus with 28 people in the room and 365 days in a year we get one pair of people having birthday on the same

Analysis Using Indicator Variable: Now we will use indicator variable to calculate an approximately simpler solution to this problem i.e. birthday paradox. We define, for each pair of people (i , j) in the room out of total people k, let us define an indicator variable: Xij indicator random variable, for 1< i < j < k so, Xij = I (person i and j having the same birthday) = Now applying expectancy on both side of the above equation, we have E(Xij) = Pr (person i and j having the same birthday) = 1/n (as calculated previously) So, for the total, say X be the random variable that counts total number of pairs having same birthday. X = ∑_(i=1)^k ∑_(j=i+1)^k Xij Taking expectations on both sides, we get E (X )=E ( ∑_(i=1)^k ∑_(j=i+1)^k Xij ) = ( ∑_(i=1)^k ∑_(j=i+1)^k E (Xij) Solving the summation mathematically we get E (X )= k(k-1)/2n If k(k-1) > 2n than the number of expected pairs having birthday on the same day will may be greater or equal to one. Solving k(k-1) > 2n we get: r = √2n + 1 Now, for example if we have n=365 and k=28 we get by using equation k(k-1)/2n => 28(27)/2(365) =1.0368 Thus with 28 people in the room and 365 days in a year we get one pair of people having birthday on the same

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