The data for the joint probability mass function of X and Y (two different measu

veudeje

veudeje

Answered question

2021-11-19

The table below contains the information for the joint probability mass function of X and Y (two different measurement systems).
a) Plot and calculate the marginal distributions of X and Y
b) Select one of the Y values from the table and find the conditional probability mass function of X for that Y value you have selected and plot it. 
c) Show whether X and Y are independent or not. 
d) Calculate the covariance of (X,Y) i.e. Cov(X,Y). 
f(x,y)1234100.0500.10200.10.10.050.05300.0500.150400.10.150.050.05

Answer & Explanation

Liek1993

Liek1993

Beginner2021-11-20Added 13 answers

Step 1 
Given, 
The table below contains the information for the joint probability mass function of X and Y (two different measurement systems).
f(x,y)1234100.0500.10200.10.10.050.05300.0500.150400.10.150.050.05 
Step 2 
a) Plot and calculate the marginal distributions of X and Y.
f(x,y)1234Total100.0500.100.15200.10.10.050.050.3300.0500.1500.2400.10.150.050.050.35Total0.30.250.350.11 
The marginal distribution of X: 
P(X=x)=y(x=x,y=y) 
X10203040P(X=x)0.150.30.20.35 
The marginal distribution of Y: 
P(Y=y)=xP(x=x,y=y) 
Y1234P(Y=y)0.30.250.350.1 
Step 3 
b) To determine X's conditional probability given a value of (Y=1) 
P(X=xY=y)=P(X=x,Y=y)P(Y=y) 
X10203040P(X=x|Y=1)0.16700.2860 
The conditional distribution is plotted: 
 
Step 4 
c) Show whether X and Y are independent or not. 
If the variable X and Y are independent that satisfies, 
P(X=xY=y)=P(X=x) 
or 
P(X=xY=y)=P(Y=y) 
Let take (X=1) 
P(X=10)=P(10, 1)+P(10, 2)+P(10, 3)+P(10, 4) 
=0.05+0+0.1+0 
=0.15 
P(X=10Y=y) 
Hence X and Y are not independent random variables 
Step 5 
d) Calculate the covariance of (X,Y) i.e. Cov(X,Y). 
Cov(x,y)=E(xy)E(x)E(y) 
Where, 
 

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