Question

Consider the probability distribution of the discrete random vector [Χ,Y ] where Χ represents the
number of orders for chickens in August at neighbouring supermarket and Y represents the number
of orders in September. The joint distribution is shown in the following table:
6
Χ
Y 100 200 300 400 500
100 0.06 0.05 0.05 0.01 0.01
200 0.07 0.05 0.01 0.01 0.01
300 0.05 0.10 0.10 0.05 0.05
400 0.05 0.02 0.01 0.01 0.03
500 0.05 0.06 0.05 0.01 0.03
(a) Find the probability that Χ ≥100 and Y ≥100
(b) Find the marginal distribution of Χ?
(c) Find the marginal distribution of Y ?
(d) Find the expected sales for August i.e. E(X )
(e) Find the expected sales for September i.e. Ε(Y )
(f) Find the conditional distribution of Y Χ = 500
(g) Find Ρ(Y ≥100 Χ = 500)
(h) Calculate the correlation coefficient of Χ and Y . (4 Marks

Answer 1

GIVEN THAT :-

According to the question we have that ,

'Χ' represents the number of orders chickens in August at neighbouring supermarket.

'Y' represents the number of orders in September.

now finding the questions below :-

TO FIND :-a)Find the probability that Χ ≥100 and Y ≥100?

now we have that

i)P(X100) ii)P(Y100)

now the solution is

i)P(X100) =0.05 + 0.01+ 0.10+ 0.01+ 0.05 + 0.01 + 0.01 + 0.05 + 0.01+ 0.01 + 0.01 + 0.01 + 0.05 + 0.03 + 0.03 (sum of the values in X,100)

there fore , P(X100)=0.44

ii)P(Y100)=0.05 + 0.10 +0.10 + 0.05 + 0.05 + 0.05 + 0.02 + 0.01 + 0.01 + 0.03 + 0.05 +0.06 +0.05 + 0.01 +0.03 = 0.67 (sumof the values in 100,Y)

there fore ,P(Y100)=0.67

TO FIND :-(b) Find the marginal distribution of Χ?

Now there fore , we have that

100 200 300 400 500 total

0.06 0.05 0.05 0.01 0.01 0.18

0.07 0.05 0.01 0.01 0.01 0.15

0.05 0.10 0.10 0.05 0.05 0.35

0.05 0.02 0.01 0.01 0.03 0.12

0.05 0.06 0.05 0.01 0.03 0.20

there fore grand total of total is 1.00

TO FIND :-c)Find the marginal distribution of Y ?

same as the marginal distribution of x

now

100 0.06 0.05 0.05 0.01 0.01

200 0.07 0.05 0.01 0.01 0.01

300 0.05 0.10 0.10 0.05 0.05

400 0.05 0.02 0.01 0.01 0.03

500 0.05 0.06 0.05 0.01 0.03

total 0.028 0.028 0.22 0.09 0.13

there fore grand total of total is 1.00

TO FIND :-e) Find the expected sales for September i.e. Ε(Y )

now finding the expected sales

E(Y)=100*0.06+200*0.15+300*0.35+400*0.12+55*0.2

E(Y)=301

TO FIND :-f) Find the conditional distribution of Y/ Χ = 500?

now having this as

conditional distribution of Y/X=500

=0.20/0.13

conditional distribution of Y/X =1.54

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