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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

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t-Distribution

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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

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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

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-3 -2 -1 0 1 2 3

5

2

The t-distribution probability density function for various values of

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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

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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

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Chi-Squared Distribution

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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

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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

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•Mean or Expected Value

2

• Standard Deviation

The Chi-Squared Model - Properties

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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

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The Chi-Squared Model - Properties

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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

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F-Distribution

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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

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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

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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

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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

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•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

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Let x(ν1, ν2) denote x with ν1 and ν2 degrees of freedom, then

12211 ,

1,

x

x

The F-Distribution - Properties

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If Y ~ F6,11,

find:

(a) P(Y < 3.09)

(b) y’ for which P(Y > y’) = 0.01

The F-Distribution – Example