FRM Part I · FRM Exam Part I · Multivariate Random Variables
The joint density of continuous variables X and Y is f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, and zero elsewhere. What is P(Y ≤ 0.5 | X = 0.5)?
The conditional probability is 0.375. The marginal density of X at 0.5 is 1, so the conditional density of Y is 0.5 + y, and integrating this from 0 to 0.5 gives 0.25 plus 0.125, which is 0.375.
- A0.125
- B0.375Correct
- C0.500
- D0.625
Explanation
The marginal density is f_X(x)=∫0^1 (x+y)dy = x+0.5, so f_X(0.5)=1. The conditional density is f(y|0.5)=(0.5+y)/1. Integrating from 0 to 0.5 gives 0.25+0.125=0.375. The value 0.125 is the joint probability P(X≤0.5,Y≤0.5), and 0.625 is the complement.
Did you get it right without looking?
One question tells you little. A timed set on Multivariate Random Variables shows your real accuracy, how long you take and where you lose marks.
More Multivariate Random Variables questions
- X takes values 1, 2 and 3. Given X=1, Y has mean 4; given X=2, mean 6; given X=3, mean 10. The marginal probabilities of X are P(1)=0.20, P(…
- X takes values 1 or 2 and Y takes values 10 or 20 with joint probabilities: P(1,10)=0.30, P(1,20)=0.20, P(2,10)=0.10, P(2,20)=0.40. What is …
- The joint probability distribution of X (0 or 1) and Y (0 or 1) is: P(0,0)=0.30, P(0,1)=0.20, P(1,0)=0.10, P(1,1)=0.40. What is P(Y=1 | X=1)…
- Two discrete variables X and Y have the joint pmf: P(0,0)=0.20, P(0,1)=0.20, P(1,0)=0.30, P(1,1)=0.30. Which statement is correct?
- Which feature distinguishes a multivariate Student's t distribution from a multivariate normal distribution with the same correlation matrix…
- A portfolio holds 60% in asset X and 40% in asset Y. The returns are jointly normal with standard deviations of 10% for X and 20% for Y, and…