Cum Distribution Function !

[Chintan]

Hi David !

I took this question from the sample exam for FRM 2007. However, I'm not able to solve this problem. Can you please guide me as to how to go about it ?

Thanks.

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26. Let Z be a standard normal random variable. An event X is defined to happen if either z
takes a value between -1 and 1 or z takes anyvalue greater than 1.5. What is the
probability of event X happening if N(1) = 0.8413, N(0.5) = 0.6915 and N(-1.5) = 0.0668,
where N() is the cumulative distribution function of a standard normal variable?

a. 0.083
b. 0.2166
c. 0.6826
d. 0.7494

The answer is :- d (0.7494)

 

[Telon]

26. Let Z be a standard normal random variable. An event X is defined to happen if either z
takes a value between -1 and 1 or z takes anyvalue greater than 1.5. What is the
probability of event X happening if N(1) = 0.8413, N(0.5) = 0.6915 and N(-1.5) = 0.0668,
where N() is the cumulative distribution function of a standard normal variable?

a. 0.083
b. 0.2166
c. 0.6826
d. 0.7494

The answer is :- d (0.7494)


P(-1 <=z<= 1) or P(z>1.5). This means P(-1 <=z<= 1) + P(z>1.5)

Try and draw the normal distribution diagrams that depict the two situations.

P(-1 <=z<= 1) = N(1) - N(-1) = N(1) - [1-N(1)]
= 2*N(1) - 1
= (2*0.8413) - 1 = 0.6826

P(z>1.5) = 1- N(1.5)
= 1- [1- N(-1.5)]
= N(-1.5)
= 0.0668

Therefore P(-1 <=z<= 1) + P(z>1.5)

0.6826 + 0.0668 = 0.7494

Hope this helps.

 

[David Harper CFA FRM]

I can't do better than Telon's explanation! Since Telon did the hard part, I drew the below because I totally agree that visualizing these ("try and draw") is the way to go. First get a picture of the area under the curve you are being asked for.

Then you also need to know that N(-1) is the same as 1 - N(1). That might be the hard part. That's true only because the normal is symmetric...David

http://learn.bionicturtle.com/images/forum/normalBrushed.jpg

 

[rahulv]

N(0)=0.5

so the area between 1 and 0 is
N(1)-N(0) =0.8413 -0.5 =0.3413

same area is between -1 and 0

so area between 1 and -1 is ---2x0.3413=0.6826

Therefore P(-1 <=z<= 1) + P(z>1.5)

0.6826 + 0.0668 = 0.7494