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Mathematics (maths) - Two Dimensional Random Variables - Important Short Objective Questions and Answers: Two Dimensional Random Variables

**Two Dimensional Random Variables**

**1.Define
Two-dimensional Random variables.**

Let S be the sample space associated
with a random experiment E. Let X=X(S) and Y=Y(S) be two functions each
assigning a real number to each s∈S.
Then (X,Y) is called a two dimensional random variable.

**2.The following table gives the joint
probability distribution of X and Y. Find the mariginal density functions of X
and Y.**

Y /X 1 2 3

1 0.1 0.1 0.2

2 0.2 0.3 0.1

**3.** If f (x, y)= kxye^{−}^{(x2}** **^{+}** **^{y}^{2}** **^{)} , x ≥ , y0 ≥ 0 is the joint pdf, find k .

5. The joint p.m.f of (X,Y) is given by P(x,
y) = k(2x + 3y), x = ,1,;02y = ,13,.2.Find the marginal probability
distribution of X.

**Answer:**

**6. If X and Y are independent RVs with
variances 8and 5.find the variance of 3X+4Y. Answer:**

Given Var(X)=8 and Var(Y)=5

**To find: **var(3X-4Y)**
**

We know that Var(aX - bY)=a^{2}Var(X)+b^{2}Var(Y)
var(3X- 4Y)=3^{2}Var(X)+4^{2}Var(Y) =(9)(8)+(16)(5)=152

**7.Find the value of k
if ***f*** **(*x*,**
***y*)** **=** ***k*(1−**
***x*)(1−**
***y*)**
for**0** **<** ***x*,**
***y*** **<** **1**
is to be joint density function. **

**Answer:**

**8. If X and Y are random variables
having the joint p.d.f**

**11**.** The joint
p.d.f of two random variables X and Y is given by**

**13. If the joint pdf of (X,Y) is f (x, y) = 6e ^{−2x−3y} , x ≥ , y0 ≥ ,
find0 the conditional density of Y given
X. **

**Answer: **

Given f (x, y)
= 6e^{−2x−3y} , x ≥ , y0 ≥ ,
0

The Marginal
p.d.f of X:

14.**Find the probability distribution
of (X+Y) given the bivariate distribution of (X,Y).**

**15. The joint p.d.f of (X,Y) is given by
***f *(*x*,* y*)*
*=* *6*e*^{−}^{(x}^{+}* ^{y}*

**Answer:**

∞ ⇒* f *(*x*)*
f *(*y*)* *=* f *(*x*,*
y*)

X and Y are independent.

**16. The joint p.d.f of a bivariate R.V
(X,Y) is given by**

**17**.**
Define Co – Variance:**

If X and Y are two two r.v.s then co –
variance between them is defined as Cov (X, Y) = E {X – E (X)} {Y – E (Y)}

(ie)
Cov (X, Y) = E (XY) – E(X) E (Y)

**18.****State
the properties of Co – variance; **

1. If
X and Y are two independent variables, then Cov (X,Y) = 0. But the

Converse need not be true

2.Cov (aX, bY) = ab Cov (X,Y)

3.Cov (X + a,Y +b) = Cov (X,Y)

5.Cov(aX+b,cY+d)=acCov(X,Y)

6.Cov
(X+Y,Z) = Cov ( X,Z) + Cov (Y,Z)

7. *Cov*(*aX* +
*bY*,*cX* + *dY*) =
*ac**σ* _{X}^{2} + *bd**σ*_{Y}^{2} + (*ad* +
*bc*)*Cov*(*X* ,*Y*) *where**σ** _{X} *

**19.Show that Cov(aX+b,cY+d)=acCov(X,Y)
Answer:**

Take
U= aX+b and V= cY+d

Then E(U)=aE(X)+b and E(V)= cE(Y)+d
U-E(U)= a[X-E(X)] and V-E(V)=c[Y-E(Y)]

Cov(aX+b,cY+d)=
Cov(U,V)=E[{U-E(U)}{V-E(V)}] = E[a{X-E(X)} c{Y-E(Y)}] =ac
E[{X-E(X)}{Y-E(Y)}]=acCov(X,Y)

**20.If X&Y are independent R.V’s
,what are the values of Var(X _{1}+X_{2})and Var(X_{1}-X_{2})
Answer:**

Var(X_{1} ±
X_{2} =) Var(X_{1} +)
Var(X_{2} )(Since X andY are independent RV then Var(aX ±
*b*X) = *a*^{2}Var(X) +
*b*^{2}Var(X) )

**21.****If
Y _{1}& Y_{2} are independent R.V’s ,then covariance (Y_{1},Y_{2})=0.Is
the converse of the above statement true?Justify your answer. **

**Answer: **

The converse is not true . Consider

X
- N(0,1)and Y = X^{2} sin *ceX* −
*N*( 1,),0

*E*(*X
*)* *=* *; 0*E*(*X
*^{3}* *)* *=*
E*(*XY*)*
*=*
*0sin*
ce all odd moments vanish*.

∴cov(*XY*)
=
*E*(*XY*) − *E*(*X* )*E*(*Y*)
=
*E*(*X* ^{3} ) −
*E*(*X* )*E*(*Y*) =
0

∴cov(*XY*)
=
0 *but X* &*Y areindependent*

**22.Show that **cov^{2}**
**(*X*** **,*Y*)** **≤**
**var(*X***
**) var(*Y*)

**Answer:**

cov(*X* ,*Y*) =
*E*(*XY*) − *E*(*X* )*E*(*Y*)
We know that [*E*(*XY*)]^{2}
≤
*E*(*X* ^{2} )*E*(*Y* ^{2} )

cov^{2}
(*X* ,*Y*) = [*E*(*XY*)]^{2}
+
[*E*(*X*
)]^{2}
[*E*(*Y*)]^{2}
−
2*E*(*XY*)*E*(*X* )*E*(*Y*)

≤ *E*(*X
*)^{2}* E*(*Y*)^{2}* *+*
*[*E*(*X
*)]^{2}*
*[*E*(*Y*)]^{2}*
*−*
*2*E*(*XY*)*E*(*X
*)*E*(*Y*)* *

≤ *E*(*X
*)^{2}* E*(*Y*)^{2}* *+*
*[*E*(*X
*)]^{2}*
*[*E*(*Y*)]^{2}*
*−*
E*(*X
*^{2}* *)*E*(*Y*)^{2}* *−*
E*(*Y
*^{2}* *)*E*(*X *)^{2}* *

= {*E*(*X* ^{2} ) − [*E*(*X* )]^{2} }{*E*(*Y* ^{2} ) − [*E*(*Y*)]^{2} }≤ var(*X* )
var(*Y*)

**23.If X and Y are independent random
variable find covariance between X+Y and X-Y. Answer:**

cov[*X*
+
*Y*, *X* − *Y* ]=
*E*[(*X*
+
*Y*)(*X* − *Y*)]−
[*E*(*X*
+
*Y*)*E*(*X* − *Y*)]

= *E*[*X
*^{2}* *]* *−*
E*[*Y
*^{2}* *]* *−*
*[*E*(*X
*)]^{2}*
*+*
*[*E*(*Y*)]^{2}*
*

= var(*X*
) −
var(*Y*)

**24.****X
and Y are independent random variables with variances 2 and 3.Find the variance
3X+4Y. **

**Answer:**

Given
var(X) = 2, var(Y) = 3

We
know that var(aX+Y) = a^{2}var(X)
+ var(Y)

And var(aX+bY) = a^{2}var(X) + b^{2}var(Y)

var(3X+4Y)
= 3^{2}var(X) + 4^{2}var(Y) = 9(2) + 16(3) = 66

**25.
Define correlation**

The correlation between two RVs X and Y
is defined as

**26.
Define uncorrelated**

Two RVs are uncorrelated with each
other, if the correlation between X and Y is equal to the product of their means.
i.e.,E[XY]=E[X].E[Y]

33.**
State the equations of the two regression lines. what is the angle between
them? Answer**:** **

Regression lines:

**34. ****The
regression lines between two random variables X and Y is given by **3*X*
+
*Y* = 10 **and**3*X* +
4*Y* = 12 **.Find the correlation between X
and Y.**

**Answer:**

**35.****Distinguish
between correlation and regression. **

**Answer: **

By correlation we mean the casual relationship
between two or more variables. By regression we mean the average relationship between two or more variables.

**36. State the Central Limit Theorem.**

standard normal distribution as *n* → ∞ provided
the m.g.f of *x _{i}* exist.

**37.****The
lifetime of a certain kind of electric bulb may be considered as a RV with mean
1200 hours and S.D 250 hours. Find the probability that the average life time
of exceeds 1250 hours using central limit theorem. **

**Solution: **

Let X denote the life time of the 60 bulbs.

Then µ = E(X)= 1200 hrs. and Var(X)=(S.D)^{2}=
σ^{2}=(250)^{2}hrs.

**38.Joint
probability distribution of (X, Y)**

Let (X,Y) be a two dimensional discrete
random variable. Let P(X=x_{i},Y=y_{j})=p_{ij}. p_{ij}
is called the probability function of (X,Y) or joint probability distribution.
If the following conditions are satisfied

1.p_{ij}
≥
0 for all i and j

The set of triples (x_{i},y_{j},
p_{ij}) i=1,2,3……. And j=1,2,3……is called the Joint probability distribution
of (X,Y)

**39.Joint
probability density function**

If (X,Y) is a two-dimensional continuous RV such
that

Then
f(x,y) is called the joint pdf of (X,Y) provided the following conditions
satisfied.

*f *(*x*,* y*)*
*≥*
*0*
for all *(*x*,* y*)∈(−∞,∞)

**40.Joint
cumulative distribution function (joint cdf)**

If (X,Y) is a two dimensional RV then *F*(*x*,
*y*) = *P*(*X* ≤
*x*,*Y* ≤ *y*) is called
joint cdf of (X,Y) In the discrete case,

**41.Marginal probability
distribution(Discrete case )**

Let (X,Y) be a two
dimensional discrete RV and p_{ij}=P(X=x_{i},Y=y_{j})
then

is
called the Marginal probability function.

The collection of pairs
{x_{i},p_{i*}} is called the Marginal probability distribution
of X.

then
the collection of pairs {x_{i},p_{*j}} is called the Marginal probability
distribution of Y.

**42.Marginal density function (Continuous
case )**

Let f(x,y) be the joint
pdf of a continuous two dimensional RV(X,Y).The marginal density

**43.Conditional
probability function**

If
p_{ij}=P(X=x_{i},Y=y_{j}) is the Joint probability
function of a two dimensional discrete RV(X,Y)

then the conditional probability function X given
Y=y_{j} is defined by

The conditional probability function Y given X=x_{i}
is defined by

**44.Conditional
density function **

Let f(x,y) be the joint pdf of a continuous two
dimensional RV(X,Y).Then the Conditional
density function of X given Y=y is defined by f (X Y) = fXY (x, y) /fY (y) ,where f(y)=marginal

**45.Define statistical properties**

Two jointly distributed RVs X and Y are
statistical independent of each other if and only if the joint probability
density function equals the product of the two marginal probability density
function

i.e.,
f(x,y)=f(x).f(y)

**46.The joint p.d.f of (X,Y) is given by ***f***
**(*x*,**
***y*)** **=** ***e*^{−}^{(x}^{+}**
**^{y}^{)}** **0** **≤**
***x*,**
***y*** **≤ ∞** .Are X and Y are**

**independent? **

**Answer**:**
**Marginal densities:

X
and Y are independent since f(x,y)=f(x).f(y)

**47.Define Co – Variance :**

If X and Y are two two r.v.s then co –
variance between them is defined as Cov (X, Y) = E {X – E (X)} {Y – E (Y)}

(ie) Cov (X, Y) = E (XY) – E(X) E (Y)

**48.
**** State the properties of Co – variance; **

1. If
X and Y are two independent variables, then Cov (X,Y) = 0. But the

Converse need not be true

2.Cov (aX, bY) = ab Cov (X,Y)

3.Cov (X + a,Y +b) = Cov (X,Y)

**49.Show that Cov(aX+b,cY+d)=acCov(X,Y)
Answer:**

Take
U= aX+b and V= cY+d

Then E(U)=aE(X)+b and E(V)= cE(Y)+d
U-E(U)= a[X-E(X)] and V-E(V)=c[Y-E(Y)]

Cov(aX+b,cY+d)= Cov(U,V)=E[{U-E(U)}{V-E(V)}] =
E[a{X-E(X)} c{Y-E(Y)}] =ac E[{X-E(X)}{Y-E(Y)}]=acCov(X,Y)

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