Question

2)

There are 4 people at a party. Consider the random variable X=’number of people having the same birthday’ (match only

month, N=12).

a. Define the sample space S and the associated probability function (pf). Plot pf.

b. Plot the associated exceedance probability function G(x)=Pr[X> x];

c. Compute mean and standard deviation of X.

Answer 1

solution! of : no. of persons having same Birth month. we have. 4 men. So to tal. no. of ways, their Birthdays on be distributed. 12x12x12x12=(1234 X can take values 0, 2, 3 and 4 So Si{0,2,3,4} Now P[x=0]= P[All B'days are different] ! 5. 4 days, assanged in 12 places ! (13)4 = 0.573 P[x=>] ; Plato B'days falling in one mouth] 128 A (12) 4 (1) 4 and two uncommon = AP[two common months]+[one common month month] £3c₂x 12 p2 j 4 x 12P2 = 0.401 (12)* (12)4. - + - Woov p[x=3] = P[ any 3 śday in a month] - AC₃ x 12 P2 = 0.025 P[x=47= p [All B'days in a common month]

pleti x pcx) | P[xxx] 1 x. p(x) , & pia) | 1 0.573 0.427 | 0 0 0.026 0.8 lobo o.org 0.001 0.075 0.225 •ool 1 0.004 20.016 &=0.879 E=1841 ) 0.401 odmt means Ex)E bepac); 0.879. var (X) = {x?pcro-secx;' - 1.841-(0.879) = 1.068 -- Std. Dev. (x): V vostr) = 17.068 = 1.033 (Ares). Graph of pdf Plot Excedence probi 0.5 0-973) 70401) 0.427) P[x= x 0.17 400310.001) 3 4 10.028) 2 10.00l)