Suppose X is a random variable with p

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Expectation and Variance

Suppose X is a random variable with p.m.f. p(x) and the values that it can possibly take on are in the set S.

The expected value (or mean) of X denoted by E(X) or m_X (or more commonly, just m ) is defined as

E(X) = m = å_xÎSx× P(X=x) = å_xÎSx× p(x)

Similarly, the expected value of some arbitrary function f(X) of the random variable X is defined as

E[f(X)] = m_f(X)= å_xÎSf(x)× P(X=x) = å_xÎSf(x)× p(x)

Some Rules of Expected Value

Assume that a and b are constants)

E(a) = a and E(b)=b.
E(aX) = aE(X) = am
E(aX+b) = E(aX) + E(b) = aE(X) + b = am+ b

The variance of X is denoted by V(X) or s²_X (or just s²) and is defined as

V(X) = E[(X-m )²] = å_xÎS(x-m )^2×p(x).

The standard deviation of X, denoted by s is equal to Ös²

A Shortcut for V(X): V(X) = s²= E(X²) - [E(X)]²= å_xÎSx^2×p(x) - m²

Some Rules of Variance

Assume that a and b are constants)

V(a) = V(b) = 0
V(aX) = a^2×V(X) = a²s²
V(aX+b) = V(aX) + V(b) = a²V(X) + 0 = a²V(X) = a²s²

(S.D. of aX+b is thus abs(a)×s )

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Previous: Discrete Random Variables: Basic Concepts

Jayant Rajgopal