1 answer

Example: A coin is tossed twice. Let X denote the number of head on the first...

Question:

Example: A coin is tossed twice. Let X denote the number of head on the first toss and Y denote the total number of heads on the 2tosses. Construct the join probability mass function of X and Y is given below and answer the following questions.

f(x, y)

x

Row Total

0   

1

y

0

1

2

Column Total

(a) Find P(X = 0,Y <= 1)

(b) Find P(X + Y = 2)

(c) Find P(Y ≤ 1)

(d) Find the conditional probability P (X = 0 | Y ≤ 1)


Answers

f(x,y) x Row toatl
0 1
y 0 1/4 0 1/4
1 1/4 1/4 1/2
2 0 1/4 1/4
column total 1/2 1/2

as there are four possible outcomes,

\left \{ HH(X=1,Y=2) ,HT(X=1,Y=1),TH(X=0,Y=1),TT(X=0,Y=0)\right \}

note that,

P(X=0,Y=2)=0=P(X=1,Y=0)

a)

P(X=0,Y\leq 1)

=P(X=0,Y=0)+P(X=0,Y=1)

=1/4+1/4

=1/2

b)

P(X+Y=2)=P(Y=2-X)

when, \quad X=0,Y=2\quad and \quad when \quad X=1,Y=1

\implies P(X+Y=2)=P(X=0,Y=0)+P(X=0,Y=1)

=0+1/4

=1/4

c)

P(Y\leq 1)

=P(Y=0,X=0,1)+P(Y=1,X=0,1)

=1/4+1/2

=3/4

d)

P(X=0|Y\leq 1 )

=\frac{P(X=0,Y\leq 1 )}{P(Y\leq 1)}

=\frac{1/2}{3/4}

=\frac{2}{3}

.

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