Respuesta :
Answer:
Step-by-step explanation:
Given that two random variables X and Y are independent. Each has a binomial distribution with success probability 0.4 and 2 trials.
When x and y are independent joint probability would be product of individual probabilities
pdf of X
X is Binom (2,0.4)
and Y is Binomi (2,0.4)
Hence joint distribution of XY would be
P(X=x, Y=y) =[tex]2Cx (0.4)^x (0.6)^{2-x} *2Cy (0.4)^y (0.6)^{2-y}[/tex]
for x=0,1,2 and y =0,1,2
b) Joint probability using table
PDF of X is
X 0 1 2
p 0.36 0.48 0.16
and same for Y also
Joint prob would be
X Y 0 1 2
0 0.1296 0.1728 0.0576
1 0.1728 0.2304 0.0768
2 0.0576 0.0768 0.0256
Joint probability distribution function are used to represent the probability of multiply variables
The joint probability distribution function is [tex]f(x,y) = ^2C_x *0.4^x * 0.6^{2- x} *^2C_y * 0.4^y * 0.6^{2- y}[/tex]
The given parameters are:
[tex]p = 0.4[/tex] --- the probability of success
[tex]n = 2[/tex] ----the number of trials
The joint probability distribution function f(x,y) is calculated as:
[tex]f(x,y) = ^nC_x * p^x * (1 -p)^{n- x} *^nC_y * p^y * (1 -p)^{n- y}\\[/tex]
So, we have:
[tex]f(x,y) = ^2C_x *0.4^x * (1 -0.4)^{2- x} *^2C_y * 0.4^y * (1 -0.4)^{2- y}[/tex]
Evaluate the differences
[tex]f(x,y) = ^2C_x *0.4^x * 0.6^{2- x} *^2C_y * 0.4^y * 0.6^{2- y}[/tex]
The above represents the joint probability distribution function f(x,y)
When x = 0, y = 0;
We have:
[tex]f(0,0) = 0.130[/tex]
When x = 0, y = 1;
We have:
[tex]f(0,1) = 0.173[/tex]
When x = 0, y = 2;
We have:
[tex]f(0,2) = 0.058[/tex]
When x = 1, y = 0;
We have:
[tex]f(1,0) = 0.173[/tex]
When x = 1, y = 1;
We have:
[tex]f(1,1) = 0.230[/tex]
When x = 1, y = 2;
We have:
[tex]f(1,2) = 0.077[/tex]
When x = 2, y = 0;
We have:
[tex]f(2,0) = 0.058[/tex]
When x = 2, y = 1;
We have:
[tex]f(2,1) = 0.077[/tex]
When x = 2, y = 2;
We have:
[tex]f(2,2) = 0.026[/tex]
So, the joint probability as a table is:
X /Y 0 1 2
0 0.1296 0.1728 0.0576
1 0.1728 0.2304 0.0768
2 0.0576 0.0768 0.0256
Read more about probability at:
https://brainly.com/question/25870256
