Question

Two fair cubical dice are thrown: one is red and one is blue. The random variable

M represents the score on the red die minus the score on the blue die.

a. Find the distribution of M.

b. Write down E(M).

c. Find Var (M)

Answer 1

a) M={-5,-4,-3,-2,-1,0,1,2,3,4,5}

$P(M=-5)=P(M=5)={\frac 1 {36}}$

$P(M=-4)=P(M=4)={\frac 2 {36}}={\frac 1 {18}}$

$P(M=-3)=P(M=3)={\frac 3 {36}}={\frac 1 {12}}$

$P(M=-2)=P(M=2)={\frac 4 {36}}={\frac 1 9}$

$P(M=-1)=P(M=1)={\frac 5 {36}}$

$P(M=0)={\frac 6 {36}}={\frac 1 6}$

b) Since the distribution is symmetrical about M=0, then mean of this distribution $E(M)=0$

c) $Var(X)=E(X^2)-E^2(X)=E(X^2)=2*5^2*{\frac 1 {36}}+2*4^2*{\frac 2 {36}}+2*3^2*{\frac 3 {36}}+2*2^2*{\frac 4 {36}}+2*1^2*{\frac 5 {36}}+2*0^2*{\frac 6 {36}}={\frac {110} {36}}={\frac {55} {18}}$