Probability & Statistics
MATH F113
Semester I 2022-2023
Dr. Yasmeen Akhtar
BITS Pilani K K Birla Goa Campus
(Figure and data source: Text book, Reference book)
Example
Let X and Y be two continuous random variables where the support of X is
[0, 1] and the support of Y is also [0, 1]. If the joint cdf of X and Y i.e., the cdf
of bivariate rv (X, Y) is FX,Y (x, y) = xy then find the joint pdf of X and Y.
Expected Values of functions of jointly distributed random variables
Proposition: Let X and Y be jointly distributed rv’s with pmf p(x, y) or
pdf f (x, y) according to whether the variables are discrete or continuous.
Then the expected value of a function h(X, Y), denoted by E[h(X, Y)]
or µh(X,Y) , is given by
(
  h(x, y)p(x, y), if X and Y are discrete
E[h(X, Y)] = R •x Ry •
• • h(x, y)f (x, y) dx dy if X and Y are continuous.
Proof. Let Z = h(X, Y), then pmf of Z is
pZ (z) = P(Z = z) = P(h(X, Y) = z) = ÂÂ p(x, y)
(x,y):h(x,y)=z
  h(x, y)p(x, y) =  Â h(x, y)p(x, y)
x y z (x,y):h(x,y)=z
=Â ÂÂ z p(x, y)
z (x,y):h(x,y)=z
= Âz ÂÂ p(x, y)
z (x,y):h(x,y)=z
= Âz P(h(X, Y) = z)
z
= Âz P(Z = z)
z
= E[Z] = E[h(X, Y)].
Expectation of Sums of Random Variables
Suppose E[X] and E[Y] both are finite. Then,
E[X + Y] = E[X] + E[Y]
Similarly, if a and b are numerical constants then
E(aX + bY) = aE(X) + bE(Y)
Proof. Let h(X, Y) = aX + bY. Then,
E[aX + bY] = Â Â(ax + by)p(x, y)
x y
= Â Â ax p(x, y) + Â Â by p(x, y)
x y x y
= a  x pX (x) + b  y pY (y)
x y
= aE[X] + bE[Y]
Expectation of Product of Independent Random Variables
Suppose X and Y are independent. Then,
E[XY] = E[X]E[Y]
Proof.
E[XY] = Â Â xy p(x, y)
x y
= Â Â xy pX (x) pY (y)
x y
= Â x(Â y pY (y)) pX (x)
x y
= E[Y] Â x pX (x)
x
E[XY] = E[X]E[Y]
Covariance
The covariance between two rv’s X and Y is
Cov(X, Y) = E[(X µX )(Y µY )]
(
  (x µX )(y µY )p(x, y) X, Y discrete
= R •x Ry •
• • (x µX )(y µY )f (x, y) dx dy X, Y continuous
Cov(X, Y) = E[XY Y µX XµY + µX µY ]
= E(XY) E[Y]µX E[X]µY + µX µY
= E(XY) µY µX µX µY + µX µY
Cov(X, Y)= E(XY) µX µY = E(XY) E(X)E(Y)
Covariance
• If X and Y are independent then Cov(X, Y) = 0.
• The covariance Cov(X, Y) is the expected product of deviations of the
two variables from their respective mean values.
• Cov(X, X) = E[(X µX )2 ]= V(X)
• Cov(X, Y) = Cov(Y, X)
• Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z)
• Cov(cX, Y) = c Cov(X, Y)
• Cov(a + bX, Y) = b Cov(X, Y)
Variance of Sums of Random Variables
If X and Y are random variables and a and b are numerical constants
then
V(aX + bY) = a2 V(X) + 2abCov(X, Y) + b2 V(Y)
If X and Y are independent then V(aX + bY) = a2 V(X) + b2 V(Y)
Covariance to measure the directional relationship
Fig: p(x, y) = 1/10 for each of ten pairs corresponding to indicated points: (a) positive covariance; (b) negative
covariance; (c) covariance near zero
Example
A nut company markets cans of deluxe mixed nuts containing almonds,
cashews, and peanuts. The net weight of each can is exactly 1 lb where the
weight contribution of each type of nut is random. Let X = weight of
almonds in a selected can, and Y = weight of cashews in a selected can
1 What is the region of positive density?
Suppose the joint pdf for (X, Y) is
(
24xy, 0 x 1, 0 y 1, x + y 1
f (x, y) =
0 otherwise.
2 What is probability that almonds and cashews together make up at
most 50% of the can?
3 Are X and Y independent r.v’s? Find the Cov(X, Y).
Covariance Drawback
• The computed value of covariance depends critically on the units of
measurement. Ideally, the choice of units should have no effect on a
measure of strength of relationship.
Let X = Policy 1 deductible amount
Y = Policy 2 deductible amount
p(x, y) y p(x, y) y
0 100 200 0 1 2
x 100 .20 .10 .20 x 1 .20 .10 .20
250 .05 .15 .30 2.5 .05 .15 .30
Correlation
The correlation coefficient of X and Y, denoted by Corr(X, Y), rX,Y ,
or just r, is defined by
Cov(X, Y)
rX,Y =
sX sY
Properties:
1. For any two rv’s X and Y, 1 r 1:
⇣ ⌘
0 V sXX + sYY = V(X)
s2
+ V(Y)
s2
+ 2Cov(X,Y)
sX sY = 2(1 + r)
X Y
) 1r
⇣ ⌘
V(X)
0 V sXX Y
sY = sX2
+ V(Y)
s2
2Cov(X,Y)
sX sY = 2(1 r)
Y
)r 1
2. The two variables are said to be uncorrelated (linearly) when r = 0.
Correlation: Properties (cont.)
3. If a and c are either both positive or both negative, then
Corr(aX + b, cY + d) = Corr(X, Y).
4. If X and Y are independent, then r = 0, but r = 0 does not imply
independence.
Example: Let X and Y be discrete rv’s with joint pmf p(x, y) = 1/4 for
(x, y) = ( 4, 1), (4, 1), (2, 2), ( 2, 2) and it is 0 o.w.
Correlation: Properties (cont.)
5. r = 1 or 1 iff Y = aX + b for some numbers a and b with a 6= 0.
6. r is actually not a completely general measure of the strength of a
relationship.
7. r is a measure of the degree of linear relationship between X and Y.
8. A value of r near 1 does not necessarily imply that increasing the
value of X causes Y to increase. It implies only that large X values are
associated with large Y values.
Correlation is NOT Causation!
