Signal, noise, and, unequal variances

[Math Is Hard]

I have two overlapping Gaussian distributions. One for probability of a noise event, one for probability of a signal event.
My noise distribution has mean = 0, variance = 1, and standard deviation = 1.
My signal distribution has mean = .80, variance = 3, and standard deviation = √3.

My criterion (λ ) is at 0.5 standard deviations above the mean on the noise distribution.

I need to calculate probability of a hit (P_H) and probability of a false alarm (P_F).

To get P_F , I use 1- P_{Correct Rejection}, or 1 – Ф(0.5), which, using a Z to % conversion table, gives me 1- 0.691 = 0.31. (In my book, Ф(Z) just means converting a Z-score to % of area under the curve.)

To get P_H, I first need to standardize the Z score for λ on the signal distribution since it has a different variance:

λ = (λ - μ_signal)/σ _signal = (0.5-0.8) / √3 = - 0.173

P_H = 1 - Ф(-0.173) or by symmetry, P_H = Ф(0.173) = 0.57. (again, using a Z to % conversion table to look up the area.)

So.. P_F = = 0.31 and P_H = 0.57.

I'm not sure if I did this right. I am shaky with unequal variances. If someone could check I'd appreciate it.

I need to sketch the overlapping distributions. Since I have standardized the z-score on the signal distribution, do I make it look identical to the noise curve (with a standard deviation of 1), or do I draw it short and wide as it is originally described?

Thanks!

 

[Almsco]

It looks like you did the calculations correctly! To sketch the overlapping distributions, draw the noise distribution with a standard deviation of 1 and then draw the signal distribution with the specified standard deviation (√3). You can make the shapes the same and just adjust the size to represent the different standard deviations. Good luck!