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Probability

SamuelMartin

New Member
Can please anyone let me know the formula applied in the solution to come up with

Prob( X < mean + 1.5*σ) = 0.9332
Prob( X < mean + 0.5*σ) = 0.6915

Prob( X < mean + 1.5*σ) - Prob( X < mean + 0.5*σ) = 0.9332 - 0.6915

Problem:

Assume that a random variable follows a normal distribution with a mean of 50 and a standard deviation of
10. What percentage of this distribution is between 55 and 65?
a. 4.56%
b. 8.96%
c. 18.15%
d. 24.17%
Answer: d.
Explanation:
Prob(mean + 0.5*σ < X < mean + 1.5*σ) = Prob( X < mean + 1.5*σ) - Prob( X < mean + 0.5*σ)
= 0.9332 - 0.6915 = 0.2417


THANK YOU VERY MUCH!!!!
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @SamuelMartin

The formula is important, it the formula for a standard (aka, unit) normal variable, typically denoted by Z. As in:
  • If μ =50 and σ = 10, then for X = 65, Z = (65-50)/10 = 1.5
  • If μ =50 and σ = 10, then for X = 55, Z = (55-50)/10 = 0.5
Then the lookups:
  • Prob(Z < 1.5 ) = 93.32% = NORM.S.DIST(1.5, CDF = true) is the area to left of +1.5 sigma
  • Prob(Z < 0.5 ) = 69.15% = NORM.S.DIST(1.5, CDF = true) is the area to left of +0.5 sigma
  • The difference between the two must be the probability of falling within +0.5 and +1.5
 

SamuelMartin

New Member
Hi David,

How can I calculate that with a calculator instead of an excel formula? With a Texas Instruments BA II Plus calculator that is one of the calculators allow in the FRM exam.

Thanks again
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @SamuelMartin You can't find the critical (lookup) Z value with the TI BA II+, but GARP is well aware of this. So, they will either just give you the Z value, or more likely (and what they are doing lately, I think they even talk about this in one of the guides), they will just give you the Z lookup table. Thanks,
 

SamuelMartin

New Member
All right, now it make sense. I was wondering how to get the values for the exam and it was driving me crazy.

Thank you so much for all your help
 
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