Determine the value of c that makes the function f(x,y) = c(x+ y) a joint probability...

Question

Question

Determine the value of c that makes the function f(x,y) = c(x+ y) a joint probability...

Determine the value of c that makes the function f(x,y) = c(x+ y) a joint probability mass function over the nine points with x= 1, 2, 3 and y = 1, 2, 3.

Determine the following:

a) P(X = 1, Y < 4)

b) P(X = 1)

c) P(Y = 2)

d) P(X < 2, Y < 2)

e) E(X), E(Y), V(X), V(Y)

f) Marginal probability distribution of the random variableX.

g) Conditional probability distribution of Y given that X =1.

h) Conditional probability distribution of X given that Y =2.

i) E(Y | X = 1)

j) ARe X and Y independent?

math Statistics-And-Probability

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

Concepts and reason

Joint probability distribution: If two or more events occur together or at same point of time, then the probability of the two events is called as joint probability. In other words, the probability of the intersection of two events is defined as joint probability. The sum of the probability of different outcomes in the probability distribution must add up to one.

Expected value: In Statistics, the long run average value obtained by repetitions for the given experiment is termed as expected value for the random variable X. Here, the random variable can be of discrete and continuous type. Moreover, as the number of repetitions is increased, the arithmetic mean of the values is approximately nearer to the expected value. The expected value of the random variable X is usually denoted by E(X).

Variance: The variation between each observation from its mean is known as variance. The greater the distance of the points from the mean, the greater is the variability and vice-versa. The variance of the random variable X is usually denoted by V(X).

Marginal probability distribution: If X and Y are two discrete random variables and let S(X,Y) be the joint probability distribution defined over the two variables, then the individual distributions of the random variables X and Y are called as marginal probability distributions of X and Y. The marginal distribution of X is denoted by fx(x) and Y is denoted by fy(y) .

Conditional probability distribution: If X and Y are two discrete random variables and let S(X,Y) be the joint probability distribution defined over the two variables. Then the distribution of random variable X given that random variable Y = y has already happened is called as the conditional probability distribution function X given Y = y. Similarly, it can be defined as for Y given X = x.

Conditional Expectation: The expected value for the conditional distribution is termed as the conditional expectation. In other words, let X be a random variable, then the expected value of the random variableX is calculated given that certain conditions has already happened.

Independent random variables: Two random variables X and Y are said to be independent random variables if the occurrence of one random variable does not affect the occurrence of another random variable. In other words, if occurring of random variable X does not influence the random variable Y occurring, then the two random variables X and Y are independent.

Fundamentals

Let X and Y are two random variables. The random variable X ranges from 1 to n, and Y ranges from 1 to m.

The joint probability distribution of random variables X and Y is,

(x,y)=f(x = x, Y = y); here x =1,2,...,n; y =1,2,...,m

The joint distribution table is given as,

y
y2
f(x,y)
f(x),y)
f(x2,y;) ... f(xmoy;)
f(x, y)... | $(xm»92)
Ym
f(x,ym)
f(x,ym)...
f(x, y)

The sum of the joint probabilitiesfor the random variables X and Y in the probability distribution is one.

That is, Ef(x = x,,Y = y,)=1; here x =1,2,...,n; y=1,2,...,m .

The formula forprobability of X equal to x and Y less than y is given as,

P(x = x,Υ <y)=ΣΣP(x = x, Y =y)

The formula for probability ofX less than x and Y less than y is given as,

P(X <x,Y < y)=
P(X = x, Y = y)
Xary

Marginal probability discrete distribution function:

For random variable X is,

For random variable Yis,

Expected value:

The formula of expected value of random variable X or E(X) is,

Ε(x)=ΣΑΡ(x = x)

Variance:

The formula of variance of X or V(x) is,

V(X)=E(X)-[E(X)] where, Ε(x2)-ΣΑΡ(Χ = )

Conditional probability distribution:

The conditional probability distribution of X given Y is,

P(X|Y = y)=-
P(X = x; Y = y). here i=1 to n
P(Y = y)

The conditional probability distribution of Y given X is,

P(Y|X = x)=-
P( X = xn Y = y;).
; here j = 1, 2,...,m
P(X = x)

Conditional expectation:

The conditional expected value of Y given Xis,

E(Y|X = x)=Xy; (Y|X = x)
P(X = X,Y = y;)
P(X = x)

Independence condition:

If random variables X and Y are independent then,

P(X = X,Y = y)=P(X = x)P(Y = y)

From the given information, the joint probability mass function of X and Y is

f(x,y)=c(x+y);
x = 1,2,3 and y=1,2,3.

ΣΣ/(α.)-ΣΣ (x + y)
=«ΣΣ(x + y)
Γ(1+1)+(1+2)+(1+3) + (2+1)+(2+2)]
“L+(243) +(3+1) +(3+ 2) + (3 +3)
= 36c

Now,

ΣΣf(x,y)=1
36c =1
c= 1

From the given information, the joint probability mass function of X and Y is

s(x,y) = 4(x+y); x=1,2,3 and y=1,2,3.

The joint probability mass function for x = 1 and y = 1 is obtained below:

(1,1)=(1+1)
136

Similarly, all the values can be obtained for different values of X and Y. Therefore, the joint probability distribution tableis given below:

Total
36
36
Total

(a)

Theprobability of the random variable X takes value 1 and Y takes value less than 4 is given below:

[P(X = 1, Y =1)+ P(X = 1, Y = 2)]
P(X = 1,Y <4)=
+P(X = 1, Y = 3)
2 3 4
E t t From joint distribution table]

-2+3+4

(b)

The marginal probability distribution function of the random variable X equal to 1is obtained below:

P(X =1)=P(X = X,Y = y)
[P(X = 1, Y = 1)+ P(X = 1, Y = 2)]
+P(X = 1, Y = 3)
2 . 3 . 4
+36 36
==+=+ From joint distribution tab

-2+3+4

(c)

The marginal probability of the random variable Y takes the value 2 is given below:

P(Y = 2) =
P(X = x, Y = y)
[P(X=1,Y = 2)+P(X = 2, Y = 2)]
*[+P(X = 3,Y = 2)
3 4 5
[From joint distribution table]

3+4+5
36
نما
- انا

(d)

The probability of the random variable X takes value less than 2 and Y takes value less than 2 is obtained below:

P(X<2,Y<2) = P(X = 1, Y =1)
[From joint distribution table]
18

(e.1)

The expected value of random variableX is obtained below:

E(X)=ExP(X= x)
9 24 45
==+*+ (From joint distribution table]
36 36 361
9+24+45
36

(e.2)

The expected value of random variable Yis obtained below:

E(Y)= Ży,P(Y = y)
1*
+2x 12
2
+3.IS
36
3
9 24 45
==+*+ [From joint distribution table]
36 36 361
9+24+45

(e.3)

The expected value of or (2x) is obtained below:

$E{x)=x*P(X=x) _9 48 135 [From joint distribiution table] +36 36 36 9+48 +135 36$

The expected value of X square or E(X) is . The expected value of X is .

The variance of X is obtained below:

v(x)= E(X?)-[E(X)]
_16 (137
315
_16 169
3 36
192-169

(e.4)

The expected value of Y square or is obtained as:

E(Y) = Žy}P(Y = y;)
From joint distribution table]
9 48 135
* 36 * 36 * 36
9+48+135
36

The expected value of X square or is . The expected value of Y is .

The required is obtained as:

V(Y)=E(Y?)-[E(Y)]
16 169
3 36
192-169
36

(f)

The marginal probability of the random variable X takes the value 1 is given below:

P(X = 1) =
P(x = 1, Y = y;)
[P(X = 1, Y =1)+ P(X = 1, Y = 2)]
*[+P(X = 1, Y = 3)
2 3 4
==+=+ From joint distribution table]
3

-2+3+4

The marginal probability of the random variable X takes the value 2 is given below:

P(x + 2) = 2 P(x = 2,= y;)
[P(X = 2, Y = 1) +P(X = 2, Y = 2)]
+P(X = 2, Y = 3)
3 4 5
za t [From joint distribution table]

3+4+5
36
نما
- انا

The marginal probability of the random variable X takes value 3 is given below:

P(X = 3) = P(X = 3, Y = y)
[P(X = 3, Y =1)+P(X = 3, Y = 2)]
[+P ( X = 3, Y = 3)
= + + [From joint distribution table]
36 36

14+5+6

(g)

The conditional probability of Y takes the value 1 given that X takes the value 1 is obtained below:

P(Y =1| X = 1) =
P(X=1nY =1)
P(X=1)
36 [From joint distribution table]
36
2 36
369

The conditional probability of Y takes the value 2 given that X takes the value 1 is obtained below:

P(Y=2|X = 1) - P(X=In Y = 2)
P(X=1)
36 [From joint distribution table]
3 36
369

The conditional probability of Y takes the value 3 given that X takes the value 1 is obtained below:

P(Y=3[ X =1)=P(X =In Y = 3)
P(X =1)
40
[From joint distribution table]
36
4 36
369

(h)

The conditional probability of X takes value 1 given that Ytakes value 2 is given below:

P(x =1 Y = 2)=P(X =lny = 2)
P(Y=2)
= 36 (From joint distribution table]
36
-3
36
*36*12

The conditional probability of X takes the value 2 given that Ytakes value the 2 is given below:

The conditional probability of X takes the value 3 given that Ytakesthe value 2 is given below:

P(X =3 Y = 2) - P(X = 3n Y = 2)
P(Y=2)
30
(From joint distribution table]
5
36
36
12
ia

(i)

The conditional expected value of random variable Y given that X takes the value 1 is obtained below:

1
2
1x 36
3 4
o [From joint distribution table]
| 36 | 36)
2x
| 36
={1x2)+(2x})+(3x4)
2+6+12
20

(j)

Consider,

The random variables X and Y are not independent.

Ans:

The value of c is .

Part a

Theprobability of the random variable X takes value 1 and Y takes value less than 4 is.

Part b

The marginal probability of the random variable X takes the value 1 is.

Part c

The marginal probability of the random variable Y takes the value 2 is.

Part d

The probability of the random variable X takes value less than 2 and Y takes value less than 2 is.

Part e.1

The expected value for random variable X is.

Part e.2

The expected value for the random variable Y is.

Part e.3

The variance of X is.

Part e.4

The variance of Y is.

Part f

The marginal probability distribution of the random variable X is given below:

Part g

The conditional probability distribution of Y given that is given below:

Part h

The conditional probability distribution of X given that is given below:

Part i

The conditional expected value of Y given that X takes the value1is.

Part j

No, the random variables X and Y are not independent.

Add a comment

Answer 2

Determine the value of c that makes the function f(x,y) = c(x+ y) a joint probability...

Homework Answers

Add Answer to:
Determine the value of c that makes the function f(x,y) = c(x+ y) a joint probability...

Post as a guest

Earn Coins

Determine the value of c that makesthe function f(x,y) = ce^(-2x-3y) a joint probability densityfunction over...

The joint probability density function for continuous random variables X and Y is given below. f...

The joint probability density function for continuous random variables X and Y is given below. f...

The joint probability density function for continuous random variables X and Y is given below. f...

Two random variables, X and Y, have joint probability density function f ( x , y...

pleaze help me fast 2. Let X and Y be discrete random variables with joint probability...

1. Let the joint probability (mass) function of X and Y be given by the following:...

The joint probability density function of the random variables X, Y, and Z is (e-(x+y+z) f(x,...

Question 4: Let X and Y be two discrete random variables with the following joint probability...

The joint probability density function of random variables X and Y is given by f(x,y) ={10xy^2...

Determine the value of c that makes the function f(x,y) = c(x+ y) a joint probability...

Homework Answers

Add Answer to: Determine the value of c that makes the function f(x,y) = c(x+ y) a joint probability...

Post as a guest

Earn Coins

Add Answer to:
Determine the value of c that makes the function f(x,y) = c(x+ y) a joint probability...