Sample Exam Question

[Deepak Chitnis]

Hi @David Harper CFA FRM CIPM, the question from Garp sample exam question (behind the book)
Assume that a random variable follows a normal distribution with mean of 80 and standard deviation of 24. What percentage of this distribution is not between 32 and 116?
A. 4.56%
B.8.96%
C.13.36%
D.18.15%
I have calculated like:32-80/24=-2 and 116-80/24=1.5. Now my question is that am I done correct step? If yes now how to look for z look up table to get the answer? And instead of -2 if there +2 how did we use the look up table? Please guide.
Thank you

 

[Dr. Jayanthi Sankaran]

Hi Deepak,

I don't know whether my answer is right or wrong. The area to the left of 2.0 (corresponding to (80 - 32)/24 = 2.0) is from the z- lookup table 1 - 0.9772 = 0.0228. The area to the right of 1.5 (corresponding to (116 - 80)/24 = 1.5) is 1 - 0.9332 = 0.0668.

Hence the area of the distribution not between 32 and 116 is 0.0228 + 0.0668 = 0.0896 = 8.96%.

Thanks!
Jayanthi

 

[Deepak Chitnis]

Hi @Jayanthi Sankaran, the anwer is correct, but I have just one query that is if there is negative value that is if there is -2 and -1.5(negative) then how will we calculate or use the z lookup for negative values. Appreciate your help. Thank you

 

[Dr. Jayanthi Sankaran]

Hi Deepak,

Glad to be of help. z lookup tables are available for negative values of z i.e. if Z is a standard normal random variable, then cumulative probabilities P(Z < z) of the standard normal distributions are available. Please refer to Appendix 1 Pg 239 of GARP's Quantitative analysis book.

When I was solving for d1 and d2 in the Black-Scholes-Merton model for solving prices of calls and puts, I used Appendix 1 extensively. It will be given to you on the exam.

Thanks!
Jayanthi

 

[Deepak Chitnis]

Hi @Jayanthi Sankaran, Thank you very much. Just wanted to ask lasr question when we compute z values are we going to add them together to get the probability? Thank you very much

 

[David Harper CFA FRM]

@Deepak Chitnis You don't necessarily add them together. You want to understand that the z value itself is a quantile (eg, z = -2 refers to two standard deviations to the left of mean which is zero in a standard normal). The lookup value itself is just a cumulative probability: the area under the curve to the left.

So, if we are trying to find the area that is NOT in between -2.0 and +1.5, we can:

• Use the typical CDF interpretation, take the area to the left of +1.5 and subtract the area to the left of -2.0: Pr(Z<+1.5) = 93.5% and Pr(Z<-2) = 2.3%; the inclusive area between [-2, 1,5, +2] = 95.3% - 2.3% ~= 91.0%. Which means the area outside this region = 100% - 91.0% ~= 9%, or Jayanthi's approach is more straightforward:
• Pr(Z<-2.0) is already outside the region so we just need to add the other tail, which is 1 - Pr(Z<+1.5) = 1 - 93.3% = 6.7%; so in this approach we add the left tail (ie, 2.3%) to the right tail (ie, 6.7%). It's important to be able to visualize this and to not rely on memorization. This whole business here needs a lookup table and a calculator, but any memorization is not a good sign because you just don't need it. I hope that helps, thanks,

 

[Deepak Chitnis]

@David Harper CFA FRM CIPM, Thank so much. That's helps a lot