Starting at a fixed time, we observe the gender of each newborn child at a certainhospital until a boy (B) is born. Let p 5 P(B), assume that successive births areindependent, and define the rv X by x 5 number of births observed. Thenp(1) 5 P(X 5 1) 5 P(B) 5 pp(2) 5 P(X 5 2) 5 P(GB) 5 P(G) ? P(B) 5 (1 2 p)pandp(3) 5 P(X 5 3) 5 P(GGB) 5 P(G) ? P(G) ? P(B) 5 (1 2 p)2pContinuing in this way, a general formula emerges:p(x) 5 5(1 2 p)x21p x 5 1, 2, 3,…0 otherwise(3.2)The parameter p can assume any value between 0 and 1. Expression (3.2) describesthe family of geometric distributions. In the gender scenario, p 5 .51 might beappropriate, but if we were looking for the first child with Rh-positive blood, then itmight be the case that p 5 .85.
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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 5 P(B), assume that successive births are
independent, and define the rv X by x 5 number of births observed. Then
p(1) 5 P(X 5 1) 5 P(B) 5 p
p(2) 5 P(X 5 2) 5 P(GB) 5 P(G) ? P(B) 5 (1 2 p)p
and
p(3) 5 P(X 5 3) 5 P(GGB) 5 P(G) ? P(G) ? P(B) 5 (1 2 p)2p
Continuing in this way, a general formula emerges:
p(x) 5 5(1 2 p)x21p x 5 1, 2, 3,…
0 otherwise
(3.2)
The parameter p can assume any value between 0 and 1. Expression (3.2) describes
the family of geometric distributions. In the gender scenario, p 5 .51 might be
appropriate, but if we were looking for the first child with Rh-positive blood, then it
might be the case that p 5 .85.
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