lec09 3.4~3.6

x 8
15
(4) P(4 X 7) b( x;15,.2) B(7;15,.2) B(3;15,.2) .348
x4
7
Example 3.32
An electronics manufacturer claims that at most 10% of its power supply units need service during the warranty period. To investigate this claim, technicians at a testing laboratory purchase 20 units and subject each one to accelerated testing to simulate use during the warranty period. Let p denote the probability that a power supply unit needs repair during the period (the proportion of all such units that need repair).The laboratory technicians must decide whether the data resulting from the experiment supports the claim that p≤0.1. Let X denote the number among the 20 sampled that need repair, so X~Bin(20, p).Consider the decision rule
X= the number of S’s among the n trials Possible values for X in an n-trial experiment are x = 0,1,2,…,n. we will often write X~Bin(n,p) to indicate that X is a binomial rv based on n trials with success probability p.
k 1 k 1
n
np C in1 p i (1 p)( n1)i np( p 1 p) n1 np
i 0
n 1
Another solution: Suppose
k nk k f ( x) (q px) C k p q x n n k 0
Solution :
6 3 3 6 P( X 3) b(3;6,.5) (.5) (.5) 20(.5) .313 3
6 x 6 x P(3 X ) b( x;6,.5) (.5) (.5) .656 x 3 x 3 x
x
P( X x) B( x; n, p) b( y; n, p)
y 0
x 0,1,2,...,n
Example 3.31 Suppose that 20% of all copies of a particular textbook
fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail the test. Then X has a binomial distribution with n=15 and p=.2. Determine the following probability
Reject the claim that p≤0.1 in favor of the conclusion that p>.10 if x≥5 and consider the claim plausible if x≤4
Solution:
The probability that the claim is rejected when p=.10 (an incorrect conclusion) is
The Binomial Random Variable and Distribution
Definition: Given a binomial experiment consisting of n trials, the
binomial random variable X associated with this experiment is defined as
n
n k k n k k 1 Derivation for two sides f ( x) np(q px)n1 kCn p q x k 0
Let x=1, then
np kC n p k q n k E ( X )
k k 0
n
Example 2: Let X have the binomial distribution with parameters n, p, determine V(X) .
P( X 5 when p .10) 1 B(4;20,.1) 1 .957 .043
The probability that the claim is not rejected when p=.20 (a different type of incorrect conclusion) is
One might think that the probability of this second type of erroneous conclusion could be made smaller by changing the cutoff value 5 in the decision rule to something else. However, although replacing 5 by a smaller number would yield a probability smaller than .630, the other probability would then increase. The only way to make both “error probabilities” small is to base the decision rule on an experiment involving many more units.
(1) the probability that at most 8 fail the test;
(2) The probability that exactly 8 fail (3) The probability that at least 8 fail (4) The probability that between 4 and 7, inclusive, fail
Example 3.30
Each of six randomly selected cola drinkers is given a glass containing cola S and one containing cola F ,The glasses are identical in appearance except for a code on the bottom to identify the cola. Suppose there is actually no tendency among cola drinkers to prefer one cola to the other. Then p=P(a selected individual prefers S)=.5, so with X= the number among the six who prefer S, X~Bin(6,.5). Determine the probability P(X=3), P(3≤X), P(X ≤1)
Solution :
(1) P( X 8) b( x;15,.2) B(8;15,.2) .009
x 0 8
15 8 158 (2) P( X 8) b(8;15,.2) (.2) (.8) .003 8
(3) P( X 8) b( x;15,.2) B(15;15,.2) B(7;15,.2) .004
The Mean and Variance of X
Proposition: If X~Bin(n,p), then E(X)=np, V(X)=np(1-p)=npq, and
X npq
Example 2: If X is a binomial rv with parameters p and n, determine the expectation.
Solution: because a binomial distribution is n Bernoulli distribution, and a Bernoulli distribution is x p(x) 0 q 1 p
E( X i ) p, E( X i 2 ) p, V ( X i ) p p2 p(1 p) pq
P( X 4 when p .) B(4;20,.2) .630
The first probability is rather small, but the second is intolerably large. When p=.20, so that the manufacturer has grossly understate the percentage of units that need service, and the stated decision rule is used, 63% of all samples will result in the manufacturer’s claim being judges plausible!