I am new to MLC: notations?

[777888]

Hi, I am new to MLC and the first thing I noticed is the numerous notations that look rather foreign to me. I am feeling extremly frustrated and I realized that I can't go any further before straightening these things out.

[note: I am using _ to represent subscripts]

1) t_p_x
Is this a function of x? a function of t? or a function of both? It seems to me that neither of the above makes sense.
If it's a function of x, why am I seeing formulas for (d/dt) t_p_x?
If it's a function of t, why am I seeing formulas for (d/dx) t_p_x?
If it's a function of both x and t, shouldn't (d/dx) t_p_x and (d/dt) t_p_x be partial derivatives?


2) In the formula for deferred mortality probability t|u_q_x, I am seeing the survival function in the form s(x+t+u), is this a function of three variables x,t, and u, i.e. f(x,t,u)=s(x+t+u) for some function f? Or are some of the variables "fixed"?


3) (Force of mortality)
I am OK with µ(x), but once again I am confused about something like µ(x+t).
Consider the formula µ(x+t)=s'(x+t)/s(x+t)
a) I don't understand the meaning of the notation µ(x+t). Is µ(x+t) a function of both x and t?
b) For s'(x+t), is it differentiating with respect to x? to t? to z=x+t?
c) For s'(x+t), is it a partial derivative?


4) (Force of mortality)
[note: once again I'm using _ to indicate subscripts]
One of the formulas says that µ_X (x+t) = µ_T(x) (t)
a) What is the meaning of putting subscripts X and T(x) to the right of µ?
b) What is the difference between µ(x+t) and µ_X (x+t)?


Please help me out. Any help would be greatly appreciated!

[777888]

I would appreciate even if you can answer only one question. Please help...

[NoMoreExams]

Weren't you answered on AO?

[FSA]

Yes, there was an answer on AO. Additionally, since you're using Bowers, see appendices 3 and 4 which provide an index as to when in the text various symbols are first used (App 3), as well as general rules for what symbols mean in various locations (App 4).

[777888]

OK, I just checked AO after posting the previous message. But I still have some troubles, feel free to drop in more comments there.

[777888]

Now I think there is something important about 3) that I don't understand.

3) Formula 1: µ(x)=-s'(x)/s(x)
Formula 2: µ(x+t)=-s'(x+t)/s(x+t)

I am OK with formula 1, but have terrible troubles understanding the exact meaning of formula 2. I have some problems understanding the meaning of s'(x+t).

Is s'(x+t) the derivative with respect to x (i.e. s'(x)=(d/dx)[s(x)] ) evatuated at x+t?
e.g.) f(x)=x^2, f '(x)=2x
f '(x+t)=2(x+t) [here the derivative itself is treated as a function and is evaluated at x+t]

OR is s'(x+t) the derivative with respect to t (i.e. (d/dt)[s(x+t)] )? If this is the case, why would they denote it simply by a prime(') without stating explictly what they are differentiating it with respect to? It's so confusing...

Could someone please explain? Any help would be appreciated!

[alekhine4149]

Now I think there is something important about 3) that I don't understand.

3) Formula 1: µ(x)=s'(x)/s(x)
Formula 2: µ(x+t)=s'(x+t)/s(x+t)

I am OK with formula 1, but have terrible troubles understanding the exact meaning of formula 2. I have some problems understanding the meaning of s'(x+t).

Is this even true? Where are your formulas from? I thought mu(x) = -s'(x)/s(x). Are you sure you aren't leaving a negative out?

[777888]

Is this even true? Where are your formulas from? I thought mu(x) = -s'(x)/s(x). Are you sure you aren't leaving a negative out?

Sorry, my bad, there should be a negative.

But the thing is that I don't understand what s'(x+t) means.

Does s'(x+t) mean "first differentiate, then evaluate at x+t", or "first evaluate at x+t, then differentiate w.r.t. t"?

This is from my study guide:

I don't understand the last equal sign. They seemingly have replaced d/dt by a prime ('). How is this step justified?
The context seems to suggest that s'(x+t) should be interpret as (d/dt)[s(x+t)], which is against the traditional way of interpreting s'(x+t) as "first differentiating, then evaluate at x+t".

These two interpretations will give two different answers , which is very troublesome in an exam situation...

Can someone please explain how s'(x+t) is being interpreted? I am very confused...

[alekhine4149]

Does s'(x+t) mean "first differentiate, then evaluate at x+t", or "first evaluate at x+t, then differentiate w.r.t. t"?

The latter. First evaluate at x+t, then differentiate w.r.t. t. I would interpret it this way because of the parentheses. Honestly, I think this is the traditional way of interpreting it.

[777888]

The latter. First evaluate at x+t, then differentiate w.r.t. t. I would interpret it this way because of the parentheses. Honestly, I think this is the traditional way of interpreting it.

But hold on, let's recall how we did these kind of things in calculus.

For example, f(x)=x^2, f '(x)=2x
f '(x+t) = ?

f '(x+t)=2(x+t) [here the derivative itself is treated as a function and is evaluated at x+t, that's how I interpret f '(x+t)]


Back to s'(x+t), if you interpret it as a derivative w.r.t. t, it would be an abuse of notation and inconsistent with the interpretation of calculus...however, the context of the derivation seems to suggest s'(x+t) means d/dt.

I am confused...