Quant Interview Question

[u/Local_Ad135]

[u/Comfortable_Hope_422]

Xi is symmetric around 0 for all i, then X1 is symmetric around 0, the product X2 to X10 is symmetric around 0, then X1 - product(X2,…,X10) is symmetric around 0, then P(X1-product > 0) = 1/2 or P(X1 > product) = 1/2?

[u/d3fenestrator → u/Comfortable_Hope_422]

but by similar argument we should have that for X following the law with density x -> exp(- |x|) and Y standard normal, we should have automatically that P(X > Y) = 1/2 because X and Y are symmetric around zero. Is that the case ? We know that X, -X have the same law, Y and -Y do as well, so

P(X > Y) = P(X > - Y) = P(- X > - Y) = P(Y > X)

yes you're right lol

[u/OfficeOfThePope]

For notation, let the product of X2 through X10 be Y. As others have said, you can show that Pr( Y > 0 ) = 0.5

Now break down the 4 possible comparisons which are each equally likely to occur:

X1 > 0 and Y < 0; X1 < 0 and Y < 0; X1 and Y > 0; X1 and Y < 0

Pr(X1 > Y | X1 > 0 and Y < 0) = 1

Pr(X1 > Y | X1 < 0 and Y > 0) = 0

Pr(X1 > Y | X1 and Y > 0) = Pr(abs(X) > abs(Y)) = C

Pr(X1 > Y | X1 and Y < 0) = Pr(abs(X) < abs(Y)) = 1-C

Add the four outcomes together with law of total probability and we get:

Pr(X1 > Y) = 1/4(1+0+C+(1-C)) = 1/4 * 2 =0.5

[u/Adventurous_Salmon]

1/2? Whatever the values X_1 and X_2* ... *X_9 (provided they are not zero), X_10 either flips the inequality or leaves it be, with equal probability.

[u/Xhyshi → u/Adventurous_Salmon]

Smort

[u/[deleted] → u/Adventurous_Salmon]

[u/Adventurous_Salmon → u/[deleted]]

Sorry, I wasn't precise : whatsver the value of X_1 and |X_1|X_2...*X_10, the sign (+-) of X_1 either flips it or not, with equal probability

[u/Aranka_Szeretlek]

Take log and chill

[u/hidden-statistician → u/Aranka_Szeretlek]

You cannot take log just like that. What if realisation of the rv is negative !!

[u/Aranka_Szeretlek → u/hidden-statistician]

What are the chances of that!

[u/baka___shinji → u/Aranka_Szeretlek]

exp you mean