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Special Continuous Probability Distributionst- Distribution
Chi-Squared Distribution F- Distribution
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS
Systems Engineering ProgramDepartment of Engineering Management, Information and Systems
2
t-Distribution
3
Definition - A random variable T is said to have the t-distribution with parameter , called degrees of freedom, if its probability density function is given by:
, - < t <
2/12
12/
2/)1( )(
t
th
where is a positive integer.
The t-Distribution
4
Remark: The distribution of T is usually called the Student-t or the t-distribution. It is customary to let
tp represent the t value above which we find an area
equal to p.
Values of T, tp,ν for which P(T > tp,ν) = p
0 tp t
p
The t-Distribution
5
-3 -2 -1 0 1 2 3
5
2
The t-distribution probability density function for various values of
6
t-table gives values of tp for various values of p and ν. The areas, p, are the column headings; the degrees of freedom, ν, are given in the left column, and the table entries are the t values.
0.4 0.3 0.2 0.15 0.1 0.05 0.025 0.02 0.015 0.01 0.0075 0.0005 0.0025 0.0005
1 0.3249 0.7265 1.3764 1.9626 3.0777 6.3137 12.7062 15.8945 21.2051 31.8210 42.4334 636.5776 127.3211 636.57762 0.2887 0.6172 1.0607 1.3862 1.8856 2.9200 4.3027 4.8487 5.6428 6.9645 8.0728 31.5998 14.0892 31.59983 0.2767 0.5844 0.9785 1.2498 1.6377 2.3534 3.1824 3.4819 3.8961 4.5407 5.0473 12.9244 7.4532 12.92444 0.2707 0.5686 0.9410 1.1896 1.5332 2.1318 2.7765 2.9985 3.2976 3.7469 4.0880 8.6101 5.5975 8.61015 0.2672 0.5594 0.9195 1.1558 1.4759 2.0150 2.5706 2.7565 3.0029 3.3649 3.6338 6.8685 4.7733 6.86856 0.2648 0.5534 0.9057 1.1342 1.4398 1.9432 2.4469 2.6122 2.8289 3.1427 3.3723 5.9587 4.3168 5.95877 0.2632 0.5491 0.8960 1.1192 1.4149 1.8946 2.3646 2.5168 2.7146 2.9979 3.2031 5.4081 4.0294 5.40818 0.2619 0.5459 0.8889 1.1081 1.3968 1.8595 2.3060 2.4490 2.6338 2.8965 3.0851 5.0414 3.8325 5.04149 0.2610 0.5435 0.8834 1.0997 1.3830 1.8331 2.2622 2.3984 2.5738 2.8214 2.9982 4.7809 3.6896 4.7809
10 0.2602 0.5415 0.8791 1.0931 1.3722 1.8125 2.2281 2.3593 2.5275 2.7638 2.9316 4.5868 3.5814 4.586811 0.2596 0.5399 0.8755 1.0877 1.3634 1.7959 2.2010 2.3281 2.4907 2.7181 2.8789 4.4369 3.4966 4.436912 0.2590 0.5386 0.8726 1.0832 1.3562 1.7823 2.1788 2.3027 2.4607 2.6810 2.8363 4.3178 3.4284 4.317813 0.2586 0.5375 0.8702 1.0795 1.3502 1.7709 2.1604 2.2816 2.4358 2.6503 2.8010 4.2209 3.3725 4.220914 0.2582 0.5366 0.8681 1.0763 1.3450 1.7613 2.1448 2.2638 2.4149 2.6245 2.7714 4.1403 3.3257 4.140315 0.2579 0.5357 0.8662 1.0735 1.3406 1.7531 2.1315 2.2485 2.3970 2.6025 2.7462 4.0728 3.2860 4.072816 0.2576 0.5350 0.8647 1.0711 1.3368 1.7459 2.1199 2.2354 2.3815 2.5835 2.7245 4.0149 3.2520 4.014917 0.2573 0.5344 0.8633 1.0690 1.3334 1.7396 2.1098 2.2238 2.3681 2.5669 2.7056 3.9651 3.2224 3.965118 0.2571 0.5338 0.8620 1.0672 1.3304 1.7341 2.1009 2.2137 2.3562 2.5524 2.6889 3.9217 3.1966 3.921719 0.2569 0.5333 0.8610 1.0655 1.3277 1.7291 2.0930 2.2047 2.3457 2.5395 2.6742 3.8833 3.1737 3.883320 0.2567 0.5329 0.8600 1.0640 1.3253 1.7247 2.0860 2.1967 2.3362 2.5280 2.6611 3.8496 3.1534 3.849621 0.2566 0.5325 0.8591 1.0627 1.3232 1.7207 2.0796 2.1894 2.3278 2.5176 2.6493 3.8193 3.1352 3.819322 0.2564 0.5321 0.8583 1.0614 1.3212 1.7171 2.0739 2.1829 2.3202 2.5083 2.6387 3.7922 3.1188 3.792223 0.2563 0.5317 0.8575 1.0603 1.3195 1.7139 2.0687 2.1770 2.3132 2.4999 2.6290 3.7676 3.1040 3.767624 0.2562 0.5314 0.8569 1.0593 1.3178 1.7109 2.0639 2.1715 2.3069 2.4922 2.6203 3.7454 3.0905 3.745425 0.2561 0.5312 0.8562 1.0584 1.3163 1.7081 2.0595 2.1666 2.3011 2.4851 2.6123 3.7251 3.0782 3.725126 0.2560 0.5309 0.8557 1.0575 1.3150 1.7056 2.0555 2.1620 2.2958 2.4786 2.6049 3.7067 3.0669 3.706727 0.2559 0.5306 0.8551 1.0567 1.3137 1.7033 2.0518 2.1578 2.2909 2.4727 2.5981 3.6895 3.0565 3.689528 0.2558 0.5304 0.8546 1.0560 1.3125 1.7011 2.0484 2.1539 2.2864 2.4671 2.5918 3.6739 3.0470 3.673929 0.2557 0.5302 0.8542 1.0553 1.3114 1.6991 2.0452 2.1503 2.2822 2.4620 2.5860 3.6595 3.0380 3.659530 0.2556 0.5300 0.8538 1.0547 1.3104 1.6973 2.0423 2.1470 2.2783 2.4573 2.5806 3.6460 3.0298 3.6460
40 0.2550 0.5286 0.8507 1.0500 1.3031 1.6839 2.0211 2.1229 2.2503 2.4233 2.5420 3.5510 2.9712 3.5510
60 0.2545 0.5272 0.8477 1.0455 1.2958 1.6706 2.0003 2.0994 2.2229 2.3901 2.5044 3.4602 2.9146 3.4602
120 0.2539 0.5258 0.8446 1.0409 1.2886 1.6576 1.9799 2.0763 2.1962 2.3578 2.4679 3.3734 2.8599 3.3734
p
•Excel
Table of t-Distribution
7
If T~t10,
find:
(a) P(0.542 < T < 2.359)
(b) P(T < -1.812)
(c) t′ for which P(T>t′) = 0.05 .
t-Distribution - Example
8
Chi-Squared Distribution
9
Definition - A random variable X is said to have the Chi-Squared distribution with parameter ν, called degrees of freedom, if the probability density function of X is:
, for x > 0
, elsewhere
where ν is a positive integer.
2
12
2/ 2/2
1 x
ex
0
)( xf
The Chi-Squared Distribution
10
Remarks:
The Chi-Squared distribution plays a vital role in statistical inference. It has considerable application in both methodology and theory. It is an important component of statistical hypothesis testing and estimation.
The Chi-Squared distribution is a special case of theGamma distribution, i.e., when = ν/2 and = 2.
The Chi-Squared Model
11
•Mean or Expected Value
2
• Standard Deviation
The Chi-Squared Model - Properties
12
It is customary to let 2p represent the value above
which we find an area of p. This is illustrated by the shaded region below.
For tabulated values of the Chi-Squared distribution see the
Chi-Squared table, which gives values of 2p for various values
of p and ν. The areas, p, are the column headings; the degrees
of freedom, ν, are given in the left column, and the table entries
are the 2 values.
• Excel
x
f(x)
p
2, p
)(1 2, pF
20
The Chi-Squared Model - Properties
13
If ,
find:
(a) P(7.261 < X < 24.996)
(b) P(X < 6.262)
(c) ’ for which P(X < ’) = 0.02
The Chi-Squared Model – Example
215~ X
14
F-Distribution
15
Definition - A random variable X is said to have the F-distribution with parameters ν1 and ν2, calleddegrees of freedom, if the probability density function is given by:
, 0 < x <
, elsewhere
2
)(
2
1
12
21
22121
21
11
)1(2/2/
)/(2/)(
x
x
The probability density function of the F-distribution depends not only on the two parameters ν1 and ν2 but also on the order in which we state them.
0
)(xh
The F-Distribution
16
Remark: The F-distribution is used in two-sample situations to draw inferences about the population variances. It is applied to many other types of problems in which the sample variances are involved. In fact, the F-distribution is called the variance ratiodistribution.
The F-Distribution - Application
17
Probability density functions for various values of ν1 and ν2
6 and 24 d.f.
6 and 10 d.f.
x0
f(x)
The F-Distribution
18
•Table: The fp is the f value above which we find an area equal to p, illustrated by the shaded area below.
For tabulated values of the F-distribution see the F table, which gives values of xp for various values of ν1 and ν2. The degrees of freedom, ν1 and ν2 are the column and row headings; and the table entries are the x values.• Excel
x
f(x)
p
px0
The F-Distribution - Properties
19
Let x(ν1, ν2) denote x with ν1 and ν2 degrees of freedom, then
12211 ,
1,
x
x
The F-Distribution - Properties
20
If Y ~ F6,11,
find:
(a) P(Y < 3.09)
(b) y’ for which P(Y > y’) = 0.01
The F-Distribution – Example