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The F Distribution
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A random variable is said to have the F distribution (named after R. A. Fisher) with v1 and v2 degrees of freedom if its density function is given by
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(45)
Percentile values of the F distribution for v1, v2 degrees of freedom are denoted by Fp,v1,v2, or briefly Fp if v1, v2 are understood. For a table giving such values in the case where p 0.95 and p 0.99, see Appendix F. The mean and variance are given, respectively, by m v2 v2 2 (v2 2) and s2 2v2(v1 v2 2 v1(v2 4)(v2 2) 2)2 (v2 4) (46)
The distribution has a unique mode at the value umode a v1 v1 2 ba v2 v2 2 b (v1 2) (47)
CHAPTER 4 Special Probability Distributions
The following theorems are important in later work.
Theorem 4-7 Let V1 and V2 be independent random variables that are chi-square distributed with v1 and v2 degrees of freedom, respectively. Then the random variable V V1 >v1 V2 >v2 (48)
has the F distribution with v1 and v2 degrees of freedom. Theorem 4-8 F1
p,v2,v1
1 Fp,v1,v2
Relationships Among Chi-Square, t, and F Distributions
Theorem 4-9 Theorem 4-10 F1
p,1,v
t2 1
(p>2), v
Fp,v,`
x2 p,v v
The Bivariate Normal Distribution
A generalization of the normal distribution to two continuous random variables X and Y is given by the joint density function f (x, y) 1 2ps1s2 21 exp e ca x m1 2 s1 b 2ra x m1 y m2 s1 b a s2 b a y m2 2 s2 b d ^2(1 r2) f (49)
where ` x ` , ` y ` ; 1, 2 are the means of X and Y; 1, 2 are the standard deviations of X and Y; and is the correlation coefficient between X and Y. We often refer to (49) as the bivariate normal distribution. 0 is necessary for independence of the random variables (see For any joint distribution the condition Theorem 3-15). In the case of (49) this condition is also sufficient (see Problem 4.51).
Miscellaneous Distributions
In the distributions listed below, the constants , , a, b, . . . are taken as positive unless otherwise stated. The characteristic function ( ) is obtained from the moment generating function, where given, by letting t i . 1. GEOMETRIC DISTRIBUTION. f (x) m 1 p P(X x) s2 pqx q p2
x M(t)
l, 2, . . . pet 1 qet
The random variable X represents the number of Bernoulli trials up to and including that in which the first success occurs. Here p is the probability of success in a single trial. 2. PASCAL S OR NEGATIVE BINOMIAL DISTRIBUTION. f (x) m P(X r P x) a s2 x r 1 r x bpq 1 rq p2
r, r
1,
M(t)
r pet b t 1 qe
The random variable X represents the number of Bernoulli trials up to and including that in which the rth success occurs. The special case r 1 gives the geometric distribution.
CHAPTER 4 Special Probability Distributions
3. EXPONENTIAL DISTRIBUTION. f (x) m 4. WEIBULL DISTRIBUTION. f (x) m a
e s2
ae ax 0 1 a2
x x M(t)
0 0 a a t
abxb 1e 0 a
x x a1
0 0 2 b b x x 8 pb a 0 0
1 b b e 2
2>b c
1 bd b
5. MAXWELL DISTRIBUTION. f (x) m 22>pa3>2x2e 0 s2
ax2>2
2 A pa
SOLVED PROBLEMS
The binomial distribution 4.1. Find the probability that in tossing a fair coin three times, there will appear (a) 3 heads, (b) 2 tails and 1 head, (c) at least 1 head, (d) not more than 1 tail.
Method 1 Let H denote heads and T denote tails, and suppose that we designate HTH, for example, to mean head on first toss, tail on second toss, and then head on third toss. Since 2 possibilities (head or tail) can occur on each toss, there are a total of (2)(2)(2) 8 possible outcomes, i.e., sample points, in the sample space. These are HHH, HHT, HTH, HTT, TTH, THH, THT, TTT For a fair coin these are assigned equal probabilities of 1>8 each. Therefore, (a) P(3 heads) P(HHH) 1 8 P(HTT < TTH < THT) P(HTT ) (c) P(at least 1 head) P(1, 2, or 3 heads) P(1 head) P(2 heads) P(3 heads) P(HHT < HTH < THH ) P(HHT ) P(HTH ) P(HHH ) P(THH ) P(HHH ) 1 8 7 8 7 8 P(HTT < THT < TTH ) P(HTT ) Alternatively, P (at least 1 head) (d) P(not more than 1 tail) 1 P(no head) 1 P(TTT ) 1 P(THT ) P(TTH ) P(THT ) 1 8 1 8 1 8 3 8
(b) P(2 tails and 1 head)
P(TTH )
P(0 tails or 1 tail) P(0 tails) P(HHH) P(HHH) 4 1 8 2 P(1 tail) P(HHT < HTH < THH) P(HHT ) P(HTH) P(THH)
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