- CFA Exams
- CFA Level I Exam
- Topic 1. Quantitative Methods
- Learning Module 3. Probability Concepts
- Subject 8. Covariance Given a Joint Probability Function

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**CFA Practice Question**

The joint probability function for X and Y is as follows:

B. -0.0075

C. 0.3065

What is the covariance between X and Y?

A. -1.1015

B. -0.0075

C. 0.3065

Correct Answer: B

First compute E(XY) = 1.1 x 0.5 x 0.15 + 1.4 x 0.5 x 0.35 + 1.1 x 0.7 x 0.4 + 1.4 x 0.7 x 0.1 = 0.7335.

Then compute the expected value for each variable:

E(X) = 1.1 x (0.15 + 0.4) + 1.4 x (0.35 + 0.1) = 1.235

E(Y) = 0.5 x (0.15 + 0.35) + 0.7 x (0.4 + 0.1) = 0.6

The covariance is then:

Cov(X,Y) = 0.7335 - 1.235 x 0.6 = -0.0075

Use the shortcut formula: Cov(X, Y) = E(XY) - E(X)E(Y).

First compute E(XY) = 1.1 x 0.5 x 0.15 + 1.4 x 0.5 x 0.35 + 1.1 x 0.7 x 0.4 + 1.4 x 0.7 x 0.1 = 0.7335.

Then compute the expected value for each variable:

E(X) = 1.1 x (0.15 + 0.4) + 1.4 x (0.35 + 0.1) = 1.235

E(Y) = 0.5 x (0.15 + 0.35) + 0.7 x (0.4 + 0.1) = 0.6

The covariance is then:

Cov(X,Y) = 0.7335 - 1.235 x 0.6 = -0.0075

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**User Contributed Comments**
16

User |
Comment |
---|---|

spartan |
how do you do this problem |

sunilcfa |
Exactly the way shown in the answer |

3CCI |
how do u arrive the shortcut formula? it's not anywhere in the notes provided. is it possible to do it any other way (with better clarity?) |

Rotigga |
Without the shortcut formula: Cov=E[(X-E(X))*(Y-E(Y))] E(X)=1.1*(0.55) + 1.4*(0.45) = 1.235 E(Y)=0.5*(0.5) + 0.7(*0.5) = 0.6 Cov=(1.1-1.235)(0.5-0.6)*0.15 + (1.4-1.235)(0.5-0.6)*0.35 + (1.1-1.235)(0.7-0.6)*0.4 + (1.4-1.235)(0.7-0.6)*0.1 = -0.0075 (B is correct) |

soarer1 |
Is there a method using BAII? |

wacarinci |
little advice for everyone looking for shortcuts with calculator...start learning and workign these problems out to better understand what you are doing. the real exam is much more theory, applying theory and far less calculations. you will hardly use your calc on exam. |

hero2607 |
good advice....wacarinci |

alallstar |
Where did we learn how to calculate E(XY) ?? I see how to do it but I don't quite get why. |

FayeMulvaney |
i have to admit I don't really understand the way the table is set out. I'm sure it's simple but I just don't see it. Could somebody please explain? Thanks |

jpducros |
within the table are the values that X and Y can be. So X can be 4 different values, and Y also. What it also gives is joint probabilities, being prob that the 2 variables X and Y be one of the 4 different values at the same time. The prob results are in the heads of the columns/lines. Hope it helps. |

Jurrens |
The two numbers in the left column (1.1 and 1.4) are the X values, and the two numbers on the top row (.5 and .7) are the Y values. |

sgossett86 |
roll up your sleeves and work this problem. you kids will spend more time looking for shortcuts than it would take to just do the work. |

sgossett86 |
i got this one first try after i spent quite a bit of time thinking about it and quite a bit saying to myself "dang, i'm gonna have to do all this, huh?" but once you get the ball rolling and take it step by step it's cut and dry. |

Shaan23 |
This is a CFA exam and stop thinking....its a waste of time...look at solutions....remember how to do it and move on.. |

abrown3474 |
Thanks, Rotigga. Got it now. |

schweitzdm |
It's hard to follow the way the matrix problem is worked through here. |