MA22S6 Numerical and Data Analysis 1 2015-2016
Homework sheet 5
Due Thursday 10th of March 2015 in class
1.
a) We consider two random variables X and Y with expectation value µX and µY respectively.
Show that the covariance
Cov(X, Y ) = E[(X − µX )(Y − µY )]
can also be written as
Cov(X, Y ) = E[XY ] − µX µY .
b) Prove that that if two random variables X and Y are independent then X and Y are
uncorrelated (i.e. their covariance vanishes)
c) For 3 random variables X, Y, Z show that
V ar[X + Y + Z] = V ar[X] + V ar[Y ] + V ar[Z] + 2Cov(X, Y ) + 2Cov(X, Z) + 2Cov(Y, Z)
2. Given the joint probability mass function (ie the probability that X = x and Y = y)
pX,Y (x, y) = P (X = x, Y = y)
in the following table
y
1
2
3
1
0
1/4
0
x
2
1/4
0
1/4
3
0
1/4
0
Show that X and Y are dependent but uncorrelated.
3. The correlation coefficient ρ(X, Y ) is defined by
ρ(X, Y ) = p
Cov(X, Y )
V ar[X]V ar[Y ]
provided that V ar[X] 6= 0 and V ar[Y ] 6= 0, otherwise ρ(X, Y ) = 0.
a) Relate ρ(rX + s, tY + u) (with real numbers r, s, t, u) to ρ(X, Y ).
b) ρ takes values in [−1, 1]. In which cases do you find ρ(X, Y ) = 1, −1, respectively?
4. A dart is thrown at a disk of radius 1 around the origin in the x − y plane. The random variables
X, Y measure
p the x- and y-coordinates with joint probability density function fX,Y (x, y) = c
(c > 0) for x2 + y 2 ≤ 1 and fX,Y (x, y) = 0 otherwise (i.e. no dart misses the disk).
a) Determine the constant c.
b) Calculate the marginal probability density functions fX (x) and fY (y).
c) Are X and Y independent random variables?
d) Are X and Y correlated?
1
Lecturer: Stefan Sint, sint@maths.tcd.ie