Читать книгу Applied Biostatistics for the Health Sciences - Richard J. Rossi - Страница 66

Two events A and B are independent if and only if one of the following conditions is met:

1 The probability of event A occurring is the same whether B occurred or not. That is,

2 The probability of event B occurring is the same whether A occurred or not. That is,

When two events are not independent, they are said to be dependent events. For example, the event that an individual has lung cancer and the event an individual has smoked a pack of cigarettes a day for several years are dependent events. On the other hand, the event an individual has lung cancer and the event an individual has blonde hair would be expected to be independent events.

If the events A and B are known to be independent, then the probability that A and B both occur is simply the product of their respective probabilities. That is, when A and B are independent

Example 2.25

The genders of successive offspring of human parents are known to be independent events. If the probability of having a male offspring is 0.48, then probability of having

1 two male offspring is

2 a male followed by a female offspring is

3 a male and a female offspring is

4 five female offspring is

5 the second offspring is a male given that the first was a female is 0.48 since successive births are independent. That is, since the successive offspring are independent

When the events A and B are independent so are the events A and not B, not A and B, and not A and not B. Thus,

Example 2.26

Achondroplasia is a genetic disorder related to dwarfism caused by an abnormal gene on one of the 23 chromosome pairs. Twenty percent of the individuals having achondroplasia inherit a mutated gene from their parents. An individual inherits one chromosome from each parent. It takes only one abnormal gene from a parent to cause dwarfism and two abnormal genes can cause death. If only one parent has a mutated gene then, there is a 50% chance that a child will receive this gene and inherit achondroplasia. On the other hand, if both parents have achondroplasia, there is a 50% chance that each parent will pass on the gene. Thus, because the parents’ genes are passed on independently, there is a 25% chance that a child will inherit neither gene, a 50% chance that the child will inherit only one abnormal gene, and a 25% chance that the child will inherit two abnormal genes and be at risk of death. To see this, suppose each parent has achondroplasia and let

and

Then, A and B are independent and P(A)=P(B)=0.5, and the probability that the child inherits

1 no abnormal genes is

2 only one abnormal gene is

3 inherits two abnormal genes is

Independence plays an important role in data collection and the analysis of the observed data. In most statistical applications, it is important that the observed data values are independent of each other. That is, knowing the value of one observation does not influence the value of any other observation.