Question

1) 4 ballpoint pens are selected without replacement at random from a box that contains 2 blue pens, 3 red pens, and 5 green pens.
If X is the number of blue pens selected and Y is the number of red pens selected

a. Write the “joint probability distribution” of x and y.

b. Find P[(X, Y ) ∈ A], where A is the region
{(x, y)|x + y ≤ 2}.

c. Show that the column and row totals of a Table and give the marginal distribution of X alone and of Y alone

d. Find the conditional distribution of Y, given that X = 1; namely, f (Y|1)=?

e. Use it to determine P(Y = 0 | X = 1).

f. Show that if the random variables X and Y are statistically independent or NOT independent.

Answer 1

A box contains 2 blue pens, 3 red pens and 5 green pens.
There are a total of 2+3+5 = 10 ballpoint pens.
Let X be the random variable denoting number of blue pens selected.
Let Y be the random variable denoting number of red pens selected.
X can take values 0,1,2.
Y can take values 0,1,2,3.
A total of 4 ball pens are chosen.
The bivariate probability distribution is given by:
  
However, x+y <5

• The joint probability distribution is given by:

  			X
  	f(x,y)	0	1	2	Total
  	0	0.02381	0.095238	0.047619	0.166667
  	1	0.142857	0.285714	0.071429	0.5
  Y	2	0.142857	0.142857	0.014286	0.3
  	3	0.02381	0.009524	0	0.033333
  	Total	0.333333	0.533333	0.133333	1

  
  
  
• 
  
  
  
  
  
  
  
• It is known that the marginal distribution of X,
  
  
  It is seen that sum of the row (of Total) is 1. The sum of the column (of Total) is 1.
  And, ,
  Hence the row total and column total of the table give the marginal distribution if X alone and Y alone respectively.
  
  
• The conditional distribution of Y given X = 1.

  Y		P(X=1)
  0	0.095238	0.533333	0.178571
  1	0.285714	0.533333	0.535714
  2	0.142857	0.533333	0.267857
  3	0.009524	0.533333	0.017857
  		Total	1

  
  Note that,
  
  
  
• P(Y = 0|X=1) from the table is 0.178571.
  
  
  
• For X and Y to be statistically independant,
  
  
  However, from the table,
  
  
  
  Hence, X and Y are statistically dependant.

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