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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%
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
Thank you so much. that helped a lot

#### 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