Without loss of generality, suppose n outcomes of the Critical Quality Attribute (CQA) are normally distributed, which is denoted by $X_i \stackrel{\text{i.i.d}}{\sim} \mathcal{N}(\mu, \sigma^2)$, where i = 1, …, n, then the distributions of sample mean and standard deviation are as known: and Moreover, sample mean and sample standard deviation are independent under normal distribution assumption.
Denote the lower and upper specification limits as L and U, respectively. The prediction or tolerance interval can be expressed by where k is a specific multiplier for the interval. For example, for prediction interval, $k=t_{1-\alpha/2,n-1}\sqrt{1+\frac{1}{n}}$.
The outcome at release can be any one of the sample, so Xrl ∼ 𝒩(μ, σ2), then the probability of passing PPQ at release should be
This probability is very easy to calculate using software, such as
pnorm() in R.
Now it is essential to obtain the bivariate joint distribution of the lower and upper prediction/tolerance interval, that is, find joint probability density function (PDF) fY1, Y2(y1, y2).
Since Y1 = X̄ − kS and Y2 = X̄ + kS, we can use another bivariate PDF fX̄, S(x, s) to calculate fY1, Y2(y1, y2) by using Jacobian transformation.
Solve X̄ and S as $x=\dfrac{y_1+y_2}{2}$ and $s=\dfrac{y_2-y_1}{2k}$, then Jacobian of the transformation is
Thus, () can be calculated as
The second equation follows from normal sample mean and standard deviation being independent.
Similarly, we can obtain the PDF of sample standard deviation fS(s). By (), let $v= \dfrac{(n-1)s^2}{\sigma^2}$, then Jacobian of the transformation is Thus,
Plug () in (), we can get the final results. where $\bar{X} \sim \mathcal{N}(\mu,
\frac{\sigma^2}{n})$ and V ∼ χ2(n − 1).
Then this quantity can be easily calculated by software, such as
functions dnorm(), dchisq() and
integrate() in R.
We can also calculate the probability of passing m PPQ batches, then under the assumption of independence and similar expected performance across batches, the probability will be Pr (Passing m batches ) = {Pr (Passing a Single PPQ Batch)}m