16.322 Stochastic Estimation and Control, Fall 2004 Prof. VanderⅤelde distributed -but for some purposes we may choose to regard some collection of he simple events as success, and employ the binomial distribution to study such ple, in the man re of transits al is drawn. The resistivity of a crystal may b regarded as a random variable taking values over a continuous range, but for the purposes of quality control in the manufacturing process, the occurance of any resistivity value within the specified tolerance may be regarded as success and all other values as failure. From this point of view, the experiment of measuring e resistivity of a crystal has only two outcomes How many units must be produced to yield P=0.95(for example)that 100 will be within specifications. Or, for a given yield, how many units must be manufactured to achieve a P=0.95 of getting 100"in-spec"items(not sequentially evaluated) 2. Determination of the binomial coefficient and binomial distribution The probability of any specified arrangement of k successes and n-k failures in n independent trials is p q where p is the probability of success on any one trial and q=l-p is the probability of failure. The probability of exactly k successes in n trials is then p q"- times the number of arrangements in which exactly k successes occur. This is due to the fact that the different arrangements are mutually exclusive so their probabilities add The probability of exactly k successes in n trials where p is the probability of success on each individual trial is P(k)=p(1-p)y- q The number s called the binomial coefficient since it is exactly the coefficient of the k" term in the binomial expansion of(a+b) (a+b) 3. Useful relations in dealing with binomial coefficients and factorials Use the convention that for integral n, =0 for k<0 and for k>n. Also use the conⅴ entions that Page 6 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde distributed – but for some purposes we may choose to regard some collection of the simple events as success, and employ the binomial distribution to study such successes. For example, in the manufacture of transistors, the resistivity of each crystal is measured after the crystal is drawn. The resistivity of a crystal may be regarded as a random variable taking values over a continuous range, but for the purposes of quality control in the manufacturing process, the occurance of any resistivity value within the specified tolerance may be regarded as success and all other values as failure. From this point of view, the experiment of measuring the resistivity of a crystal has only two outcomes. How many units must be produced to yield P=0.95 (for example) that 100 will be within specifications. Or, for a given yield, how many units must be manufactured to achieve a P=0.95 of getting 100 “in-spec” items (not sequentially evaluated). 2. Determination of the binomial coefficient and binomial distribution The probability of any specified arrangement of k successes and n-k failures in n k nk independent trials is p q − where p is the probability of success on any one trial and q=1-p is the probability of failure. The probability of exactly k successes in n k n k trials is then p q − times the number of arrangements in which exactly k successes occur. This is due to the fact that the different arrangements are mutually exclusive so their probabilities add. The probability of exactly k successes in n trials where p is the probability of success on each individual trial is ⎛ ⎞ n n ( ) = ⎜ ⎟ p k (1− p) n−k = ⎛ ⎞ k n−k k Pk p q k ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ n The number ⎜ ⎟is called the binomial coefficient since it is exactly the coefficient k⎝ ⎠ of the k th term in the binomial expansion of (a b) n + . n n n ⎛ ⎞ k n−k (a b + ) = ∑⎜ ⎟a b k =0 k⎝ ⎠ 3. Useful relations in dealing with binomial coefficients and factorials ⎛ ⎞ n Use the convention that for integral n, ⎜ ⎟ = 0 for k<0 and for k>n. Also use the k⎝ ⎠ conventions that Page 6 of 10