Using the properties of expected value, prove the shortcut rule for the population covariance: Cov(X,Y)=E(XY)-E(X)E(Y)
Added by Donna P.
Step 1
Step 1: Recall the definition of the population covariance between two random variables \(X\) and \(Y\): \[ \mathrm{Cov}(X,Y) = E\big[(X - E(X))(Y - E(Y))\big] \] Show more…
Show all steps
Close
Your feedback will help us improve your experience
Cheng Zhang and 54 other Intro Stats / AP Statistics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Using the properties of summation, prove the shortcut rule for the sample covariance: Cov(X,Y) = XY - XY. HINT: In this class, the sample covariance between X and Y is defined as: Cov(X,Y) = ̑̑̑(Xi - X)(Yi - Y). Upload a file containing your proof.
Cheng Z.
Madhur L.
Suppose X and Y are random variables with means ̄μX and μY and variances σX^2 and σY^2 respectively. Use the definition of covariance (Cov(X,Y) = E[(X - μX)(Y - μY)]) to prove the following: Cov(X,Y) = E[XY] - E[X]E[Y]. Cov(aX + b,cY + d) = acCov(X,Y), where a, b, c, and d are constants. Recall that if X and Y are independent, then Var(X + Y) = Var(X) + Var(Y). A more general formula that applies whether or not X and Y are independent is Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y). Prove this formula.
Adi S.
Recommended Textbooks
Elementary Statistics a Step by Step Approach
The Practice of Statistics for AP
Introductory Statistics
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD