Suppose two dice are rolled. Let X be the random variable measuring the sum of the two numbers rolled.
(a) Find the probability mass function for X.
(b) Find the expected value E(X).
(c) Find the variance V(X).

The expected value E(X) is 7 and the value of variance of X is 5.8333.

Two dice are rolled. Let X be the random variable measuring the sum of the two numbers rolled.

a) The probability of mass function is obtained below:

The possible outcomes in each of the dice are 1 to 6. Therefore, the possible outcomes when two dice is rolled is 36

The sample space, s for fair dice (red die and blue die) is given below:

N(s) = 36

From the given information, two dice are rolled let X be the random variable measuring the sum of the two numbers rolled.

b) The expected value is calculated below:

The probability mass function of X is,

The required mean is,

E(x) = ∑xP(X=x)

[tex]=[2[/tex]×[tex]\frac{1}{36}+3[/tex]×[tex]\frac{2}{36}[/tex]+....+11×[tex]\frac{2}{36}+12[/tex]×[tex](\frac{1}{36} )[/tex]]

=[0.0556+0.1667+0.3333+0.5556+0.8333+1.1111+1.000+0.8333+0.6111+0.3333

=7

C) The variance V(X) is calculated below:

The probability mass function of X is,

The required variance is,

V(x)= ∑[tex](X-x)^{2}P(x) = 5.8333[/tex]

Therefore, the expected value E(X) is 7 and the value of variance of X is 5.8333.

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