Expected values E(X)

Understanding Expected Values E(X)

Definition

• Expected value of a random variable is the weighted average of all possible values that this random variable can take on.
• It is also known as the mean or first moment of the distribution.
• Expected values E(X) provide a measure of the central tendency of the probability distribution of a random variable.

Mathematics of Expected Values

• The expected value of a discrete random variable X is calculated as: E(X) = Σ[x * P(X = x)], where P(X = x) is the probability density function.
• For a continuous random variable, the expected value is calculated as: E(X) = ∫x * f(x) dx over the entire range of X, where f(x) is the probability density function.

Properties

• Linearity: E(aX + b) = aE(X) + b, where a and b are constants. This means that the expected value operator is linear.
• The expected value of a constant is the constant itself: E(c) = c.
• E(X + Y) = E(X) + E(Y) irrespective of whether X and Y are independent or not.

Significance and Application

• Expected values are used to calculate other statistical measures like variance and standard deviation.
• In real-life scenarios, it’s used to predict future occurrences, often in the context of insurance, finance, and risk assessment.

Limitations

• Expected value may not always provide a good sense of the ‘typical’ value if the distribution is skewed as it is influenced by outliers.
• Knowing the expected value alone is not enough to understand the variability or uncertainty associated with the variable’s possible outcomes.