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# Probability Distribution, Mean and Variance

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The following problem develop the concept of determining the probability distribution of a random variable and its mean and variance.
A fair coin is tossed three times. Describe the sample space Ω.
Let X be random variable that denotes the number of heads on the first toss. Describe the probability frequency
distribution of X. Find the mean E(X) and variance Var(X).
Let Y be another random variable that counts the total number of heads. Describe the probability frequency distribution
of Y , and evaluate E(Y) and Var(Y ).
Describe the frequency distribution of the ordered pair Z = (X, Y ). Then evaluate E(Z) and Var(Z).
Describe the frequency distribution of the product W = XY . Then evaluate E(W) and Var(W).

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The solution illustrates the concept of determining the probability distribution of a random variable and its mean and variance.

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The sample space Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Let X be random variable that denotes the number of heads on the first toss.
Then X can take the values 0 and 1 with probabilities P(X = 0) = 1/2, and
P(X = 1) = 1/2.
Thus, the probability frequency distribution of X is
x 0 1
P(X = x) 1/2 1/2
Now,
E(X) = ∑xP(X = x) = (0)(1/2) + (1)(1/2) = 1/2 = 0.5
E(X^2) = ∑(x^2)P(X = x) = (0^2)(1/2) + (1^2)(1/2) = 1/2 = 0.5
Var(X) = E(X^2) - [E(X)]2 = (1/2) - (1/2)^2 = (1/2) - (1/4) = 1/4 = 0.25
Let Y be random variable that counts the total number of heads.
Then Y can take the values 0, 1, 2 and 3 with ...

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