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MathidiaMathematics Academy
Probability II
Lesson 32 of 75
Easy practice

Multivariate distributions

Objective. Calculate marginals, conditionals and covariance.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Multivariate distributions describe several random variables jointly through a joint density f(x,y), from which marginal densities are found by integrating out the other variable: f_X(x) = ∫ f(x,y) dy. Conditional densities are f(y|x) = f(x,y)/f_X(x), and covariance measures linear association via Cov(X,Y) = E[XY] − E[X]E[Y]. For a joint density f(x,y) = 1 on the unit square, X and Y are independent since f(x,y) factorises as f_X(x)f_Y(y) = 1·1, giving zero covariance.

Worked example

For the joint density f(x,y) = 6xy² on 0 ≤ x,y ≤ 1, find the marginal distribution of X.

y) = 6xy² on 0 ≤ x
Illustrated card showing y) = 6xy² on 0 ≤ x
  1. 1Integrate over y: f_X(x) = ∫0^1 6xy² dy
  2. 2Antiderivative of y² is y³/3, giving 6x·(1/3)
  3. 3Simplify 6x/3 to 2x
  4. 4State the marginal density for 0 ≤ x ≤ 1

Answer: f_X(x) = 2x

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Common mistake. Students often assume zero covariance implies independence, but this holds only in special cases like joint normality; in general variables can be dependent yet uncorrelated. Always check the joint density factorises before claiming independence.

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