Page 51 - DMTH404_STATISTICS

Unit 3: Conditional Probability and Independence Baye’s Theorem

Notice that the sum of the probabilities of the eight points in  is Notes

which is as it should be.
Summarising the discussion so far, consider two random experiments  and  with sample
l 2
spaces S and S , respectively. Let u , u . . . be the points of S and let v , v . . ., be the points of S .
1 2 1 2 1 1 2 2
Suppose p , p , . . ,a nd q , q ,. . . are the associated probabilities, i.e., P(u ) = p and P(v) = q, with
1 2 l 2 i i j j
p , q  0  p  1,  q  1. We say that  and  are independent experiments if the events “first
i j i j 1 2
i j
outcome is u ” and the event “second outcome is v ”, are independent,
i j
i.e., if the assignment of probabilities on the product space S1 x S2 is such that
P{(u, v)} = P(u ) P(v) = p q
j j i j i j
This assignment is a valid assignment because P((u , v))  0 and
i j
 P(u ,v ) =  p q i
i
j
i
i j i j
=  p i q  1.
j
i j
where the sums are taken over all values of i and j.
Can we extend these concepts to the case of n (n > 2) random experiments?
Let us denote the n random experiments by  ,  , . . . ,  . Let S . S , . . . , S be the corresponding
1 2 n 1 2 n
sample spaces. Let P(x ) denote the probability assigned to the outcome x of the random
j j
experiment . We say that  . . . ,  are independent experiments, if the assignment of probabilities
j 1 n
on the product space S × S × . . . × S is such that
1 2 n
P{(x , x , . . ., x )} = P(x ) P(x ) . . . P(x ).
1 2 n 1 2 n
The random experiments  . . . ,  are said to be repeated independent trials of an experiment  if
1 n
the sample space of  ,. . .  are all identical and so are the assignment of probabilities, it is in this
1 n
sense that the experiment discussed in Example 26 corresponds to 3 independent repetitions of
the experiment of inspecting a unit, where the probability of a unit being defective is P.
Before we conclude ow discussion of product spaces and repeated aials, let us revert to the case
of two independent experiments  and  with sample spaces S and S
1 2 1 2
Suppose
S = (u , u , . . . ), P(u ) = p , i  1
1 1 2 j i
S = (v , v ,... ), P(v) = q, j  1
2 1 2 j j
Let A = (u ,u ,....) and A = (v ,v ,...) be two events in S and S . Then A × A is event in
1 1 i 2 i 2 1 j 2 j 1 2 1 2
S × S and
1 2
A × A = {(u , v ) | r, s = 1, 2, . . .}.
1 2 r i s j
Under the assumption that E~ and E~ are independent, we can write

P(A1 × A2) =  P{(u , v )}
s j
r i
r s
=  p q j
i

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