Expected Value and Variance of Continuous RVs

Why Integration Replaces Summation for E(X)

For a discrete random variable, the expected value is the probability-weighted sum: E(X) = ∑ x_k · P(X = x_k). Each value x_k is weighted by how probable it is. For a continuous random variable, there are no individual probabilities — instead, the probability density f(x) tells us how densely probability is packed near each value x. The continuous analogue of the sum is the integral:

E(X) = ∫ x · f(x) dx

Interpret this geometrically: E(X) is the balance point of the pdf curve. If you could cut out the region under the pdf graph from cardboard, E(X) is the x-coordinate where it would balance on a knife edge. For symmetric distributions, E(X) equals the median and the axis of symmetry.

Variance as Average Squared Deviation

Variance measures how spread out the distribution is around its mean μ = E(X). By definition:

Var(X) = E[(X − μ)²] = ∫ (x − μ)² f(x) dx

Each squared deviation (x − μ)² is weighted by the density f(x). Large deviations far from the mean contribute heavily. The shortcut formula Var(X) = E(X²) − [E(X)]² follows from expanding the square and using linearity of integration — it is algebraically equivalent but computationally much faster.

The standard deviation SD(X) = √Var(X) is in the same units as X, making it the more interpretable measure of spread.

The Mode

The mode of a continuous distribution is the value of x where the pdf is maximum — the peak of the density curve. Find it by differentiating f(x) and setting f′(x) = 0, then confirming it is a maximum (f′′(x) < 0 or by inspection). For simple pdfs on a closed interval, also check the endpoints.

Note: for a uniform distribution (f(x) = constant), every point is equally a mode. For strictly increasing pdfs, the mode is at the right endpoint; for strictly decreasing pdfs, at the left endpoint.

The Median vs the Mean

The median m splits the distribution equally: P(X ≤ m) = 0.5. The mean E(X) is the balance point. For symmetric distributions, median = mean. For skewed distributions:

• Right-skewed: a long right tail pulls the mean to the right of the median
• Left-skewed: a long left tail pulls the mean to the left of the median

The median is more robust to extreme values and is often a better measure of the “typical” value in skewed distributions.

Linear Transformations: Effect on E(X) and Var(X)

If Y = aX + b (a linear function of X), then:

E(Y) = aE(X) + b. Multiplying X by a scales the mean by a; adding b shifts the mean by b.

Var(Y) = a²Var(X). Multiplying X by a scales the variance by a². Adding a constant b has no effect on variance — shifting all values left or right does not change how spread out they are.

SD(Y) = |a| · SD(X). Standard deviation scales by |a|, not a².