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Chapter 15, Probabilities (Normal Distribution)

N

ntalwar

Member
Hi,

I am really struggling with understanding how are the values of the probabilities obtained in questions relating to finding the probability if an accumulation of a sum will be more or less than a given value. For example, in the exam-style question 2, how is P(Z > 0.547) = 0.292? I see the value of ᶲ (0.55) = 0.70884 (which is the nearest value for x = 0.547) in the tables on page 160 of the actuarial tables.

Q&A Bank Part 4 Question 4.9 (asking if the accumulated value will be more than £230) - how is P(Z > 1.165) = 1 - 0.87799? The nearest value to 1.165 in the tables is 1.17 and ᶲ (1.17) = 0.87900.

Basically, I want to find out how do you calculate ᶲ (x) for a given x. I know that it is a complex calculation involving the erf function, but why do the course notes/Q&A Bank then have exact values in the calculations instead of just nearest values from the tables?

Also, what is the effect of changing signs within the brackets, i.e. changing < to >, or <= to >=?

Would appreciate if I can get an explanation of above? This is really troubling me as I can understand the probability distribution function of a standard normal distribution but cannot see how the course notes are calculating exact values when the web (wikipedia etc) describe calculating probabilities as somewhat impossible except with a computer program.
 
ntalwar said:
Hi,

I am really struggling with understanding how are the values of the probabilities obtained in questions relating to finding the probability if an accumulation of a sum will be more or less than a given value. For example, in the exam-style question 2, how is P(Z > 0.547) = 0.292? I see the value of ᶲ (0.55) = 0.70884 (which is the nearest value for x = 0.547) in the tables on page 160 of the actuarial tables.

Q&A Bank Part 4 Question 4.9 (asking if the accumulated value will be more than £230) - how is P(Z > 1.165) = 1 - 0.87799? The nearest value to 1.165 in the tables is 1.17 and ᶲ (1.17) = 0.87900.

Basically, I want to find out how do you calculate ᶲ (x) for a given x. I know that it is a complex calculation involving the erf function, but why do the course notes/Q&A Bank then have exact values in the calculations instead of just nearest values from the tables?

Also, what is the effect of changing signs within the brackets, i.e. changing < to >, or <= to >=?

Would appreciate if I can get an explanation of above? This is really troubling me as I can understand the probability distribution function of a standard normal distribution but cannot see how the course notes are calculating exact values when the web (wikipedia etc) describe calculating probabilities as somewhat impossible except with a computer program.
Click to expand...

We use linear interpolation on the normal tables.

P(Z>0.547) = 1 - P(Z<0.547)

From the Tables:

P(Z<0.54) = 0.70540
P(Z<0.55) = 0.70884

So:

P(Z<0.547) = P(Z<0.54) + 0.7[P(Z<0.55) - P(Z<0.54)] = 0.70781

Hence:

P(Z>0.547) = 1 - 0.70781 = 0.29219

To be honest the current CT1 examiners are OK with you using the nearest value in the Tables. However, interpolation is expected for the later CT subjects (eg CT3 and CT6).

With regard to changing signs:

P(Z<z) look up in the Tables
P(Z>z) = 1 - P(Z<z)

P(Z<-z) = P(Z>z) = 1 - P(Z<z)
P(Z>-z) = P(Z<z)

Since the normal distribution is continuous (and can take an infinite number of values) the P(Z=z)=0 hence P(Z<=z) = P(Z<z)
 
Hi John,

Many thanks for your help. I am clear on this now.

Regards
Nish
 
I was looking for this explanation.
Thank you guys.

And John, could you please tell me which all pages will be required from the Actuarial formula tables for the CT1 exam so that I can be well prepared for the exam

Thanks,
Warrior
 
Off the top of my head:

Lognormal distribution - page 14
Increasing annuity formula, accumulation factor with variable delta - page 31
Formulae for forward prices - page 45 (but using different notation to Chapter 13)
Compound interest section - pages 49-66 (up to you whether you use these or not).
Standard normal tables - pages 160, 161, 162
 
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