Cumulative probability for standard normal variable

CFA Level I Quant
#1

Hi i am confused about why cumulative Z tables for standard normal distribution mention that z value is greater than or equal to zero.

Like so P(Z less than or equal to z ) for z greater than or equal to zero…

cuz as i understand … z is number of standard deviations above or below expected value… and if its below mean… z will be negative… so why is it only for z greater than or equal to zero…

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#2

For z=1.000, is the table showing .3413 or.8413?

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#3

Because the standard normal distribution is symmetric about zero,

P(z \le Z) = P(z \ge -Z) = 1 - P(z < -Z)
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#4

Because the normal distribution curve is symmetrical, the mean is zero. Therefore, the Z table can be given by only positive values.

You can have a look at the only positive Z scores table here : https://www.sjsu.edu/faculty/gerstman/StatPrimer/positive-z.pdf

Also, no matter a Z score is negative or positive the area under the curve is the same.

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The definition here is:

  • F(-z) = P(Z < -z) = 1 - F(z)
  • P(Z > z) = 1 - F(z)
  • P(a < Z < b) = F(b) - F(a)

For instance,

P(Z < -1.33) = 9.18%
P(Z > 1.33) = 9.18%
P(Z > -1.33) = 1 - 9.18% = 0.9082

Edit: You can manually draw the curve of @S2000magician’s equation and you will understand. (The area under the curve is the same)

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#6

@breadmaker. For the one in the pic… It’s for 1 std deviation below mean… The prob is 0. 1587… For 1 std above mean… It’s. 8413…

#7

@Pyng thankyou

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#8

This, of course, isn’t true as written.

All normal distributions are symmetric about their mean; not all normal distributions have a mean of zero.

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#9

@Pyng I think he meant zero standard deviations away

#10

It was supposed to be written as If Z ~ N(0, 1), then Z is said to follow a standard normal distribution. I like when @S2000magician corrects my statements. :grin:

Thanks @yasin.

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#11

I figure that you write stuff like that just to keep me on my toes.

:wink:

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#12

I dare not, Sir. Thank you for the heads-up. :pray: :slightly_smiling_face:

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