4.2 Composite trials  (Page 4/4)

Because Bernoulli sequences are used in so many practical situations as models for success-failure trials, the probabilities $P(S_n=k)$ and $P(S_n\ge k)$ have been calculated and tabulated for a variety of combinations of the parameters $(n,p)$ . Such tables are found in most mathematical handbooks. Tables of $P(S_n=k)$ are usually given a title such as binomial distribution, individual terms . Tables of $P(S_n\ge k)$ have a designation such as binomial distribution, cumulative terms . Note, however, some tables for cumulative terms give $P(S_n\le k)$ . Care should betaken to note which convention is used.

In general, there are k or more successes in n trials iff there are not $n-k+1$ or more failures.

m-functions for binomial probabilities

Although tables are convenient for calculation, they impose serious limitations on the available parameter values, and when the values arefound in a table, they must still be entered into the problem. Fortunately, we have convenient m-functions for these distributions. When MATLABis available, it is much easier to generate the needed probabilities than to look them up in a table, and the numbers are entered directly into the MATLAB workspace. And we have greatfreedom in selection of parameter values. For example we may use n of a thousand or more, while tables are usually limited to n of 20, or at most 30. The two m-functions for calculating $P\left(A_{kn}\right)$ and $P\left(B_{kn}\right)$ are

• $P\left(A_{kn}\right)$ is calculated by y = ibinom(n,p,k) , where k is a row or column vector of integers between 0 and n . The result y is a row vector of the same size as k .
• $P\left(B_{kn}\right)$ is calculated by y = cbinom(n,p,k) , where k is a row or column vector of integers between 0 and n . The result y is a row vector of the same size as k .

Note that we have used probability $p=0.39$ . It is quite unlikely that a table will have this probability. Although we use only $n=10$ , frequently it is desirable to use values of several hundred. The m-functions work well for n up to 1000 (and even higher for small values of p or for values very near to one). Hence, there is great freedom from the limitations of tables.If a table with a specific range of values is desired, an m-procedure called binomial produces such a table. The use of large n raises the question of cumulation of errors in sums or products. The level of precision in MATLAB calculations is sufficient that such roundoff errors are well below pratical concerns.

Remark . While the m-procedure binomial is useful for constructing a table, it is usually not as convenient for problems as the m-functions ibinom or cbinom. The lattercalculate the desired values and put them directly into the MATLAB workspace .

Joint bernoulli trials

Bernoulli trials may be used to model a variety of practical problems. One such is to compare the results of two sequences of Bernoulli trials carried outindependently. The following simple example illustrates the use of MATLAB for this.

The ease and simplicity of calculation with MATLAB make it feasible to consider the effect of different values of n . Is there an optimum number of throws for Mary? Why should there be an optimum?

An alternate treatment of this problem in the unit on Independent Random Variables utilizes techniques for independent simplerandom variables.

Alternate matlab implementations

Alternate implementations of the functions for probability calculations are found in the Statistical Package available as a supplementary package. We have utilized our formulation, so that only the basic MATLAB package is needed.