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# R16.P1.T2.HULL_CH11:Topic:BIVARIATE_NORMAL_DISTRIBUTION_Eg

##### Active Member
Hi,
In Reference to R16.P1.T2.HULL_CH11:Topic:BIVARIATE_NORMAL_DISTRIBUTION_Eg :-

While Generating the Correlated Sample X(1) & X(2), why are we plugging in the -ve of e(1) & e(2) in the formula ....?   #### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@gargi.adhikari ε(1) and ε(2) are independently (ie, uncorrelated) randomly generated standard normals, N(0,1); the learning XLS is helpful here, these are each given by the same =ROUND(NORMSINV(RAND()),2). Z(1) pulls from the ε(1) without altering it, but Z(2) needs to translate ε(2) in order to generate a correlated but still random standard normal with the expression: z(2) = ρ*ε(1) + ε(2) *sqrt(1-ρ^2); in the above, z(2) = 0.80*-1.460 + -1.520*sqrt(1-0.80^2) = -2.08.

So here Z(1) = ε(1) = -1.460 but Z(2) = -2.08, which is the random normal -1.520 transformed into a correlated-to-ε(1) random standard normal of -2.08.

But Z(1) and Z(2) are now just the embedded correlated standard normals, as the final step they are scaled by their respective mean and σ; e.g., X(2) = 5 + 9*Z(2). I hope that clarifies!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@gargi.adhikari (1.460) with parens is negative, compare it to positive 0.990 (third to last) although, apologies, they aren't colored red like the other columns so it could be clearer. This page can be improved (cc: @Nicole Seaman can we task for minor update please): also the E(1) and E(2) switches to X(1) and X(2) which is not helping as the concept is already hard enough. Thanks!

##### Active Member
Thanks so much for the clarification @David Harper CFA FRM  ...thought I was missing something conceptually... ##### Active Member
@David Harper CFA FRM A follow Up question on this Topic for Clarification...
Slide # 1:
Slide #1, says Z1 & Z2 are the Independent Standard Normal Samples and we generate Correlated Samples e1 & e2 from the Independent samples Z1 & Z2 using the formula e2= rho* Z1 + Z2* SQRT(1-who^2) which is good

But Slide 2 while describing the numerical example says e1 & e2 are the Independent Standard-Normal variables...when as per Slide e1 & e2 should be correlated...
Could you please confirm if e1 & e2 are actually Correlated and generated as such..? Thanks - hope I am not missing something here... :-( Slide # 2: #### berrymucho

##### Member
Not sure whether the slides appeared back-to-back in the original material, both are correct, but the notation is different. If you pay attention in slide 1 the bivariate correlated variables are E1 and E2, with Z1 and Z2 the univariate independent random variables, where the letter Z is commonly used to refer to N(0,1) variables (think z-score!). In slide 2 the bivariate correlated variables are X1 and X2, with the univariate independent random variables designated by epsilon1 and epsilon2, where the letter epsilon (not capital E) is also commonly used to refer to N(0,1) variables in some other context (think regression residuals!). Bottom line, I don't think you're missing anything, just be mindful that the notation for some variables may use different letters. This should not be too confusing if you're clear about what the structure of the expression should look like.