When you add independent random variables, means add, variances add, and the distribution of the sum is the convolution of the individual distributions. The sum gets wider and more bell-shaped.
Convolution
If X and Y are independent discrete random variables, the PMF of Z = X + Y is the convolution: P(Z = z) = sum over k of P(X = k) P(Y = z - k). Each possible way the parts can add to z contributes to the total. This is why adding dice produces the familiar triangle-shaped distribution.
Scheme
; Convolution: P(Z = z) = sum_k P(X = k) * P(Y = z - k); Adding two dice: distribution of the sum
(define (convolve pmf-x pmf-y max-x max-y)
; Returns a list of (value . probability) for Z = X + Y
(let loop ((z 2) (result '()))
(if (> z (+ max-x max-y)) (reverse result)
(let inner ((k 1) (prob 0))
(if (> k max-x)
(loop (+ z 1) (cons (cons z prob) result))
(let ((y-val (- z k)))
(if (or (< y-val 1) (> y-val max-y))
(inner (+ k 1) prob)
(inner (+ k 1)
(+ prob (* (pmf-x k) (pmf-y y-val)))))))))))
; Fair die PMF
(define (fair-die k)
(if (and (>= k 1) (<= k 6)) (/ 1.06) 0))
(define two-dice (convolve fair-die fair-die 66))
(display "Sum of two dice:") (newline)
(for-each (lambda (pair)
(display "P(Z=") (display (car pair)) (display ") = ")
(display (cdr pair)) (newline))
two-dice)
Python
# Convolution: sum of two dicedef convolve(pmf_x, pmf_y, max_x, max_y):
result = {}
for x inrange(1, max_x + 1):
for y inrange(1, max_y + 1):
z = x + y
result[z] = result.get(z, 0) + pmf_x(x) * pmf_y(y)
return result
fair = lambda k: 1/6if1 <= k <= 6else0
two_dice = convolve(fair, fair, 6, 6)
for z insorted(two_dice):
bar = "#" * int(two_dice[z] * 100)
print(f"P(Z={z:2d}) = {two_dice[z]:.4f} {bar}")
Means add
E[X + Y] = E[X] + E[Y]. This is linearity of expectation from Ch 6, and it holds whether or not X and Y are independent. Roll two dice: E[X + Y] = 3.5 + 3.5 = 7. Roll a hundred dice: E[sum] = 350.
Scheme
; E[X + Y] = E[X] + E[Y] -- always
(define (expected-die)
(/ (+ 123456) 6.0))
(display "E[one die] = ") (display (expected-die)) (newline)
(display "E[two dice] = ") (display (* 2 (expected-die))) (newline)
(display "E[100 dice] = ") (display (* 100 (expected-die))) (newline)
; Works for any random variables, not just dice; E[height + weight] = E[height] + E[weight]; even though height and weight are correlated
(display "Linearity does not require independence.")
Variances add when independent
Var(X + Y) = Var(X) + Var(Y), but only when X and Y are independent. If they are correlated, a covariance term appears: Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X,Y). For independent variables, covariance is zero, so variances add cleanly.
Theory says: means add, variances add. Simulation confirms it. Roll two dice many times, compute the sample mean and variance of the sum, and check that they match the formulas.
Scheme
; Verify convolution properties by brute force; Enumerate all 36 outcomes of two dice
(define (all-sums)
(let loop-x ((x 1) (sums '()))
(if (> x 6) sums
(let loop-y ((y 1) (s sums))
(if (> y 6) (loop-x (+ x 1) s)
(loop-y (+ y 1) (cons (+ x y) s)))))))
(define sums (all-sums))
(define n (length sums))
(define (list-sum lst)
(if (null? lst) 0 (+ (car lst) (list-sum (cdr lst)))))
(define mean (/ (list-sum sums) n))
(define var-val
(/ (list-sum (map (lambda (s) (* (- s mean) (- s mean))) sums)) n))
(display "Enumerated mean of X+Y: ") (display (* 1.0 mean)) (newline)
(display "Expected (3.5 + 3.5): 7.0") (newline)
(display "Enumerated variance: ") (display (* 1.0 var-val)) (newline)
(display "Expected (2.917+2.917): 5.833")
Python
# Verify by enumeration
sums = [x + y for x inrange(1, 7) for y inrange(1, 7)]
mean = sum(sums) / len(sums)
var = sum((s - mean)**2for s in sums) / len(sums)
print(f"Mean of X+Y: {mean:.4f} (expected 7.0)")
print(f"Var of X+Y: {var:.4f} (expected 5.833)")
print(f"SD of X+Y: {var**0.5:.4f}")
Notation reference
Notation
Scheme
Meaning
P(Z = z) = ∑ P(X=k)P(Y=z-k)
(convolve pmf-x pmf-y ...)
Convolution of PMFs
E[X+Y] = E[X]+E[Y]
(+ mu-x mu-y)
Means always add
Var(X+Y) = Var(X)+Var(Y)
(+ var-x var-y)
Variances add (if independent)
Cov(X,Y)
E[(X-μx)(Y-μy)]
Covariance: dependence correction
Neighbors
Probability chapters
🎰 Ch 6 — expected value and variance (prerequisites for this chapter)
🎰 Ch 5 — the named distributions that get convolved here
🎰 Ch 8 — law of large numbers (why sums of many variables become predictable)
Convolution
The Handshake