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

Meant for a calculus-based path in stochastic approaches on the graduate or complex undergraduate point, this article deals a contemporary, utilized perspective.Instead of the traditional formal and mathematically rigorous technique ordinary for texts for this path, Edward Kao emphasizes the improvement of operational talents and research via various well-chosen examples.

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**Additional resources for An Introduction to Stochastic Processes **

**Example text**

N À k)! (K À k)! N! (N À K À n þ k)! ¼ n K(K À 1) Á Á Á (K À k þ 1) k N(N À 1) Á Á Á (N À k þ 1) Â (N À K)(N À K À 1) Á Á Á (N À K À n þ k þ 1) (N À k)(N À K À 1) Á Á Á (N À k À n þ k þ 1) (4:6:3) Since K ) k and N ) n, Eq. 7) can be approximated as n K Á K Á Á Á K (N À K)(N À K) Á Á Á (N À K) P{E ¼ k} ¼ N Á N ÁÁÁN k N Á N ÁÁÁN k times (4:6:4) NÀk times Approximating p ﬃ K/N and q ﬃ (N 2 K )/n in Eq. 8), we obtain n k nÀk P{E ¼ k} ﬃ pq k (4:6:5) which is a binomial distribution. 7 POISSON DISTRIBUTION Perhaps three of the most important distributions in probability theory are (1) Gaussian, (2) Poisson, and (3) binomial.

The first success occurs at the mth draw after (m 2 1) failures. Since the failure probability is q ¼ [r/(b þ r)], we have from the definition of the geometric distribution this probability is [r/(b þ r)]m21. The same result can also be obtained from the cumulative summation as follows: kÀ1 1 1 X b b X r g k, ¼ bþr b þ r k¼m b þ r k¼m # mÀ1 " 2 b r r r þ ¼ 1þ þÁÁÁ bþr bþr bþr bþr 2 3 mÀ1 mÀ1 b r 1 r 6 7 ¼ ¼ 4 r 5 bþr bþr bþr 1À bþr This geometric distribution is shown in Fig.

The problem can be stated as follows. Four dice each with 4 faces are tossed without regard to order and with replacement. Find the number of ways in which 1. This can be achieved. With n ¼ 4 and r ¼ 4 in Eq. 7), the number of ways is 4À1þ4 4 ¼ 7 7 ¼ 35 ¼ 3 4 which agrees with the total number in the table. 2. Exactly one digit is present. With n ¼ 4 and r ¼ 1, the number of ways is 4À1þ1 1 ¼ 4 ¼4 1 which agrees with the table. 3. Exactly 2À digits are present. The number of ways 2 digits can be drawn from 4 Á digits is 42 .