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A Bivariate standard normal distribution is a bivariate normal distribution whose two marginal means are zero and marginal variances are one; its remaining parameter is their correlation.

A conditional distribution describes the law of a random quantity after specifying information about another. If $Y$ takes values in a measurable space and $X$ is the conditioning variable, a regular conditional distribution is a measurable probability kernel $K(x,A)$ satisfyingfor measurable sets $A,B$. Thus $K(X,A)$ is a version of $\mathbb{P}(Y\in A\mid X)$ for every target event. On standard Borel spaces such kernels exist, and their values are determined only for ${\mathbb{P}}_X$-almost every $x$.

For discrete variables at a point of positive mass, the conditional probability mass function is the familiar joint-mass/marginal-mass ratio. For variables with a joint probability density, the analogous ratio $f_{Y\mid X=x}(y)=f_{X,Y}(x,y)/f_X(x)$ applies where $f_X(x)>0$. When $X$ is continuous, $\mathbb{P}(X=x)=0$ typically holds, so dividing event probabilities by $\mathbb{P}(X=x)$ cannot define the conditional law; the kernel or density construction supplies it instead.

For a bivariate standard normal distribution with correlation $\rho$, the conditional law $Y\mid X=x$ is normal with mean $\rho x$ and variance $1-{\rho }^2$. This shows how an observation changes prediction and remaining uncertainty while the original marginal law of $Y$ remains standard normal.

Under $H_0$, $(Z_1,Z_2)$ is a bivariate standard normal distribution with correlation $\rho =\sqrt{m_1/m_2}$. Rejection occurs either at stage 1 through $Z_1\ge u_1$, or at stage 2 through $Z_2\ge u_2$ after continuation $l_1<Z_1<u_1$. Hence the Type I error isThe final lack-of-benefit boundary $l_2$ affects acceptance, but not the probability of crossing an efficacy boundary.