Fall 2005; #14

[EaglesFan]

I'm obviously missing something but why wouldn't it just be -.87 in the numerator for (dP/du)?

Thanx

[campbell]

It's the distinction between Capital Phi and lower-case phi. The first is the cdf of the standard normal; the second is the density function.

[EaglesFan]

Thanx. Possibly the only time I've seen lower-case phi tested on an SOA exam. Of course, could be a harbinger for November......

[campbell]

You could also do a chain rule on the Phi((ln 5000 - mu)/sigma) directly, which will give you the -phi(z)/sigma answer.

[its_me]

Isn't g(x) = F(5000) here?

I haven't ever heard of upper & lower phi !

If they had to give dg/d_mu and dg/d_sigma where g(x)= F(x), how would it appear??

[its_me]

Isn't g(x) = F(5000) here?

I haven't ever heard of upper & lower phi !

If they had to give dg/d_mu and dg/d_sigma where g(x)= F(x), how would it appear??

[its_me]

:bump:
I can't seem to get the Uppercase Phi & lowercase phi. Someone please explain this to me.

[ActuaryToBe]

Upper case phi corresponds to the standard normal cdf:
i.e. Upper case phi = F(z) , where z=(x-mu)/sigma

Lower case phi corresponds to the standard normal density (I believe):
i.e. Lower case phi = [1/sqrt(2pi)]exp(-z^2/2) where z=(x-mu)/sigma

There is a difference in the notation, albeit not easy to spot in my opinion under exam conditions given the limited exposure we have to lower case phi. I spent about 30 minutes on that question because I was calculating it as upper case phi instead of lower case phi as well. I wouldn't be surprised if a lot of students missed this question simply because of the notation.

[joeshmoe1]

Isn't Var(F(x)) = (F(x)*(1-F(X)))/n? Why can't you use this to come up with the s.d.

[campbell]

Just confirming what was stated earlier: lower-case phi is the probability density function for the standard normal distribution. Upper-case phi is the cdf.

[Abraham Weishaus]

Isn't Var(F(x)) = (F(x)*(1-F(X)))/n? Why can't you use this to come up with the s.d.
You could if the empirical estimator were used -- this is the variance of the empirical estimator. The question states, however, that the maximum likelihood estimator was used.

[Abraham Weishaus]

Isn't g(x) = F(5000) here?

I haven't ever heard of upper & lower phi !

If they had to give dg/d_mu and dg/d_sigma where g(x)= F(x), how would it appear??
That is correct, so \frac{\partial g}{\partial \mu}=\frac{-\phi\left[(5000-\mu)/\sigma\right]}{\sigma}=-\frac{e^{-(5000-\mu)^2/(2\sigma^2)}}{\sqrt{2\pi}}\div\sigma, for example.