By Edward P. C.(Edward P.C. Kao) Kao

ISBN-10: 0534255183

ISBN-13: 9780534255183

Meant for a calculus-based path in stochastic tactics on the graduate or complex undergraduate point, this article bargains a latest, utilized viewpoint. rather than the normal formal and mathematically rigorous procedure traditional for texts for this direction, Edward Kao emphasizes the improvement of operational talents and research via a number of well-chosen examples.

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**Example text**

Convert the fractions Find out how many times the original denominator goes into the common denominator. Multiply the numerator by the same number. It is now possible to compare the fractions. denominator 7 12 denominator multiples of 12 multiples of 8 3, 6, 9, 12, 8, 16, 24, 32, 12, 24, 36, 40, 48, 56, 48, 60, 72, 15, 18, 21, 84, 96… 24, 27, 30… 64, 72… lowest common denominator of 3, 8, and 12 common denominators 24, 48, 72, 96… largest fraction smallest fraction 2 ×8 16 5 ×3 15 = = 3 ×8 24 8 ×3 24 original denominator goes into common denominator 8 times, so multiply both numerator and denominator by 8 original denominator goes into common denominator 3 times, so multiply both numerator and denominator by 3 7 ×2 14 = 12 ×2 24 original denominator goes into common denominator 2 times, so multiply both numerator and denominator by 2 FRACTIONS ADDING AND SUBTRACTING FRACTIONS Just like whole numbers, it is possible to add and subtract fractions.

00 6 Dividing 40 by 8 gives 5 exactly, and the division ends (terminates). 75. 0 0 60 –56 4 Above, short division was used to ﬁnd the decimal answer for the sum 8 ÷ 6. Long division can be used to achieve the same result. 0 0 8 ﬁts into 6 0 times, so write 0 here multiply 8 times 0 to get 0 First, divide 8 into 6. It goes 0 times, so put a 0 above the 6. Multiply 8 × 0, and write the result (0) under the 6. 0 0 60 add decimal point bring down a 0 divide 60 by 8 Subtract 0 from 6 to get 6, and bring down a 0.

00 6 Dividing 40 by 8 gives 5 exactly, and the division ends (terminates). 75. 0 0 60 –56 4 Above, short division was used to ﬁnd the decimal answer for the sum 8 ÷ 6. Long division can be used to achieve the same result. 0 0 8 ﬁts into 6 0 times, so write 0 here multiply 8 times 0 to get 0 First, divide 8 into 6. It goes 0 times, so put a 0 above the 6. Multiply 8 × 0, and write the result (0) under the 6. 0 0 60 add decimal point bring down a 0 divide 60 by 8 Subtract 0 from 6 to get 6, and bring down a 0.

### An Introduction to Stochastic Processes by Edward P. C.(Edward P.C. Kao) Kao

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