Que: 11 Suppose at a fixed time, we observe the gender of each new born child at a certain hospital until a boy (B) is born. Let P(B) = p, assume that successive birth are independent and let the r.v. X is defined as x= no. of births observed. Find the pmf and cdf of X.
Answers
Answer:
Starting at a fixed time, we observe the gender of each newborn child at a certain hospital until a boy (B) is born.
Let
p
=
P
(
B
)
, assume that successive births are independent, and define the rv
X
by
X
=
number of births observed. Then
p
(
1
)
=
P
(
X
=
1
)
=
P
(
B
)
=
_____
p
(
2
)
=
P
(
X
=
2
)
=
P
(
G
B
)
=
P
(
G
)
⋅
P
(
B
)
=
_____
and
p
(
3
)
=
P
(
X
=
3
)
=
P
(
G
G
B
)
=
P
(
G
)
⋅
P
(
G
)
⋅
P
(
B
)
=
_____
Continuing in this way, a general formula emerges:
P
(
x
)
=
{
(
1
−
p
)
x
−
1
p
x
=
1
,
2
,
3
,
.
.
.
0
otherwise
The parameter
p
can assume any value between _____ and _____.
The expression above describes the family of geometric distributions. In the gender example,
p
=
0.51
might be appropriate, but if we were looking for the first child with Rh-positive blood, then we might have
p
=
0.85
.
Answer:
The pmf and cdf of X is 0.85.
Step-by-step explanation:
Step : 1 Starting at a fixed time, we observe the gender of each newborn child at a certain hospital until a boy (B) is born. Let p = P(B),
assume that successive births are independent, and define the rv X by X = number of births observed.
Then,
Step : 2 p(1) = P(X = 1) = P(B) = p(2) = P{X = 2) = P(GB) = P(G) middot
P(B) = and p(3) = P(X = 3) = P(GGB) = P(G) middot P(G) middot P(B) = Continuing in this way, a general formula emerges:
P(x) = {(1 - p)^x - 1_p x = 1, 2, 3, ... 0 otherwise
Step : 3 The parameter p can assume any value between and The expression above describes the family of geometric distributions.
In the gender example, p = 0.51 might be appropriate,
but if we were looking for the first child with Rh-positive blood,
then,
we might have p = 0.85.
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