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What is the variance of random variable X whose value when two fair dice are rolled is X((i,j)) = i + j , where i and j are the numbers appearing on the first and second die respectively ?

a) 35/6

b) 49/4

c) 91/6

d) 35/12

2 Answers

2 votes
2 votes

Now, the values of X are

X= 2,3,4,5......., 12

We note the values of X and their corresponding probablities without the factor $\frac{1}{36}$

X 2 3 4 5 6 7 8 9 10 11 12
P(x) 1 2 3 4 5 6 5 4 3 2 1

Now, V(X) = E(X2) - E2(X)   ..........(1)

------------------------------------------------------------------------------------------------

We calculate the mean i.e E(X)

Now, just look at P(X) values from 1 to 6. X values for them are one greater. 

So, this is $\sum_{1}^{6}$n(n+1)

Now, for the rest of the values, if P(X)=n then X=13-n

So, this is $\sum_{1}^{5}$n(13-n)

So, E(x)

= $\sum_{1}^{6}$n(n+1) + $\sum_{1}^{5}$n(13-n)

= 7

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We also calculate E(X2)

E(X2)

= $\sum_{1}^{6}$n(n+1)2 + $\sum_{1}^{5}$n(13-n)2

=  $\frac{1974}{36}$

--------------------------------------------------------------------------------------------

Now, from (1), we get 

V(X)

= $\frac{1974}{36}$ - 72

= $\frac{35}{6}$

So, option A should be the correct answer.

edited by
1 votes
1 votes
ok .... i got it now ...

lets take x1 = random variable for outcome of first die

and x2 = random variable for outcome for second die

thus we have x = x1 + x2

now, since x1 and x2 are independent events

we have var(x) = var(x1) + var(x2)

we know var(x1)=var(x2) and finding its value is easy

final answer will be (A)

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