Question

# The random variable X can take on the values 1, 2 and 3 and the random...

The random variable X can take on the values 1, 2 and 3 and the random variable Y can take on the values 1, 3, and 4. The joint probability distribution of X and Y is given in the following table:

 Y 1 3 4 X 1 0.1 0.15 0.1 2 0.1 0.1 0.1 3 0.1 0.2

a. What value should go in the blank cell?

b. Describe in words and notation the event that has probability 0.2 in the table.

c. Calculate the marginal distribution of X and the marginal distribution of Y.

d. Are X and Y independent events? Show why or why not with calculations.

e. Calculate the conditional distribution of X given Y=1.

X and Y be the two random variable with possible values 1,2,3 and 1,3,4 respectively.

The joint probability distribution of X and Y

 Y X\Y 1 3 4 1 0.1 0.15 0.1 X 2 0.1 0.1 0.1 3 a 0.1 0.2

(a)

We have two conditions (i) and (ii)

 X\Y 1 3 4 Total 1 0.1 0.15 0.1 0.35 2 0.1 0.1 0.1 0.3 3 0.05 0.1 0.2 0.35 Total 0.25 0.35 0.4 1

We get,

a=0.05

(b)

The event that has probability 0.2 in the table means probability of X=3 and Y=4.

i.e.

(c)

The marginal probability of X is,

 X 1 2 3 Total P(x) 0.35 0.3 0.35 1

The marginal probabilty of Y is,

 Y 1 3 4 Total P(y) 0.25 0.35 0.4 1

(d)

X and Y are called independent random variables iff ,

In other words

for all i and j.

Here,

Simalarly we find all probabilitie,  condition of independecy not holds;

Here, we senen that X and Y are not independent.

(e)

The conditional distribution of X given Y=1.

 Total Y=1 0.1 0.1 0.05 0.25

The conditional probability of X given Y=1 ,

 X 1 2 3 Total P(X/Y=1) 0.4 0.4 0.2 1

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