Page 355 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes 1
of each of these is . Conditioned by the event A, i.e. that the first die shows a five, the
6
(conditional) event that the sum is nine comprises of only one sample point (5, 4) i.e. the sample
point common to both A and B. Here the conditional probability of getting a sum of nine given
1 1
that the first die shows a five is , or in symbols P (B|A) = .
6 6
• Consider a conditional event (B|A) i.e. the event B given that A has actually happend. Then for
the happening of the event (B|A), the sample space is restricted to the sample points comprising
the event A. The conditional probability P (B|A) is given by
i
P (B|A) = .
j
• These ideas are not equivalent ideas. We discuss them to bring out the difference between the
two. When the happening of one event precludes the happening of the other event, the two
events are mutually exclusive (or (disjoint). For two mutually exclusive events A and B
P (AB) = 0.
• When the happening of one event has no effect on the probability of occurrence of happening
of the other event, the two events are independent. For two independent events A and B,
P (AB) = P (A) P (B).
• Two events can be mutually exclusive and not independent. Again two events can be
independent and not mutually exclusive.
Suppose two coins are tossed. The events {H, H} ≡ A (head on both coins) and the event {T, T}
≡ B are mutually exclusive (because if A happens B cannot happen) and P (AB) = 0. But P (A)
1 1
= , P (B) = and so P (AB) = 0 ≠ P (A) P (B) and hence A and B are not independent.
4 4
•
• A sample space that consists of a finite number of sample points or a denumerably infinite
number of sample points is called a discrete sample space.
27.4 Key-Words

1. Effective sample size : The sample size needed in equal-sized groups to achieve the power when
we have groups of unequal sizes. It will generally be less than the total
number of subjects in the unequal groups.
2. Efficiency : The degree to which repeated values for a statistic cluster around the
parameter.
27.5 Review Questions

1. State and prove the multiplicative theorem of probability. How is the result modified when the
events are independents ?
2. State and prove the addition rule of probability.
3. Differentiate between the circumstances when the probabilities of two events (i) added, and (ii)
multiplied.
4. Discuss the general rule for probability. What is its form if the concerned events are mutually
exclusive ?
5. Distinguish between addition and multiplicative rule of probability.

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