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Cumulative Standard Normal Distribution

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    #31
    Re: Cumulative Standard Normal Distribution

    This horse doesn't want to die!
    I made an observation and was trying to explain it mathematically.
    No matter what the number (X) is, the max number of iterations never went over 171 !!
    Why?
    Simple: The number generated from "X^(2*j+1" will have 165 digits (including decimals)!! I don't even know what the term for this one is! But it's far above and beyond any computer calculating abilities.
    So, it's safe to just put in any number in the loop of 171 or higher.

    That said, I would like to give special thanks to Steve Andrews for his help in this thread,... and even Ted has his arterial motives, but.... thanks for jumping in.

    Comment


      #32
      Re: Cumulative Standard Normal Distribution

      Good observation, Gabe. The bottom line is that since we are working with a "standard" normal distribution, the standard deviation is 1.0. With that said, using the function for arguments such as 3 or 5 (in the examples you provided) is rarely needed and quite impractical.

      Thus, you could have different code to determine the result for arguments of 3 and above and only use the Taylor series code for argumanets below 3. That different code would only have to determine whether to return a value of 0.999998, 0.999999, or 1.000000.

      It is true that the terms in the series loop get wild for larger arguments due to those exponential powers. Regardless, it still converges to the exact amount when not subject to limitations of the computer.

      Comment


        #33
        Re: Cumulative Standard Normal Distribution

        Steve:
        I woke up this morning and said to myself:
        What am I thinking?!: The max P(probability) is 1 and the min is 0 even though neither one is practical and my intention is to test at these 2 extremes which I did and the max number of iteration was 10 for 1, and 1 (as expected) for zero.
        That settles it. Any number >10 should do.
        Settles it as far as that go... but then I had couple more observations, will post after testing.
        Thanks

        Comment


          #34
          Re: Cumulative Standard Normal Distribution

          Originally posted by G Gabriel View Post
          The max P(probability) is 1 and the min is 0 even though neither one is practical and my intention is to test at these 2 extremes which I did ...
          Keep in mind that the argument you are giving the function is not the probability but instead a random variable whose value can be anything (from minus infinity to infinity). It will normally be from -1 to 1 (within one standard deviation of the mean) about two thirds of the time. But it will almost never be less than -3 or higher than 3. For those cases, the cumulative function will be close to zero or one, respectively.

          Comment


            #35
            Re: Cumulative Standard Normal Distribution

            That would be true in Normal Distribution, but in Standard Normal Distribution with a mean of 0 the number supplied by user (actually in my case, it is supplied by another calculation) has to fall within 1 SD of zero.

            Comment


              #36
              Re: Cumulative Standard Normal Distribution

              Originally posted by G Gabriel View Post
              That would be true in Normal Distribution, but in Standard Normal Distribution with a mean of 0 the number supplied by user (actually in my case, it is supplied by another calculation) has to fall within 1 SD of zero.
              I disagree. The standard deviation is by no means the limit. A statistical observance may easily be, say, 1.5 standard deviations from the mean.

              Comment


                #37
                Re: Cumulative Standard Normal Distribution

                The definition of Standard Normal Distribution:
                A standard normal distribution is a normal distribution with mean 0 and standard deviation 1
                Isn't it?

                Comment


                  #38
                  Re: Cumulative Standard Normal Distribution

                  Originally posted by G Gabriel View Post
                  The definition of Standard Normal Distribution:
                  A standard normal distribution is a normal distribution with mean 0 and standard deviation 1
                  Isn't it?
                  Yes, it is. But having a standard deviation of 1.0 by no means indicates that the random variable can be no larger than 1.0 (nor less than -1.0). It only says that about 66% of the sampled data will fall within that range.

                  Comment


                    #39
                    Re: Cumulative Standard Normal Distribution

                    68% will fall within that range and that range is 1 SD of zero, which is what?

                    Comment


                      #40
                      Re: Cumulative Standard Normal Distribution

                      And about 95% will fall between two standard deviations of the mean (between -2 and 2 for a standard normal distribution).

                      Comment


                        #41
                        Re: Cumulative Standard Normal Distribution

                        I understand.. but we are only (in Standard ND and Standard CND) interested in 1 SD. Furthermore, any number above say 3 will result in an outcome of 1 (i.e. 100%), i.e. What are the probability of finding 1,000 or less when the mean is zero? 100%. What are the probability of finding 5 or less when the mean is zero? Same thing: 100%. The question is: what are the probability of finding a number (user's) at 1 SD from zero, or less?
                        If the number supplied by the user is > 1 SD of zero, the outcome will not change because the calculation takes into effect 1 SD.
                        If I am not mistaken, 1 SD of zero is .70 or thereabout. So any number entered by the user >.7 whether it is .70 or a million, the answer is the same.

                        Comment


                          #42
                          Re: Cumulative Standard Normal Distribution

                          Originally posted by G Gabriel View Post
                          ... but we are only (in Standard ND and Standard CND) interested in 1 SD.

                          The question is: what are the probability of finding a number (user's) at 1 SD from zero, or less?

                          If the number supplied by the user is > 1 SD of zero, the outcome will not change because the calculation takes into effect 1 SD.
                          If I am not mistaken, 1 SD of zero is .70 or thereabout. So any number entered by the user >.7 whether it is .70 or a million, the answer is the same.
                          You keep confusing the probabilities (return values) with the random variables (user-entered values). If you're "only interested in 1SD" then you are not interested in the full range of practical statistics.

                          While you may see probabilities of 100% that the entered value is 5 or less, that is only due to rounding. The true probability is closer to 99.99997%. The probabilities of a CSND theoretically never reach 100% unless rounding to a certain number of digits.

                          The probability of a value being less than 1 SD, is NORMSDIST(1) = 84.134%. But the probability of being "within one SD of zero" is NORMSDIST(1) - NORMSDIST(-1) = 68.270%.
                          Last edited by Steve Andrews; 08-22-2011, 05:51 PM.

                          Comment


                            #43
                            Re: Cumulative Standard Normal Distribution

                            Steve:
                            We going around and around with this..
                            I am not confusing probability with the user's entered number. What I am saying is, since in STANDARD CND the mean is zero and SD is one, no matter what number the user enters, if it is more than .68 (One Standard Deviation of zero) the outcome will be the same because STANDARD CND measure the area under the bell within 1 SD of zero.
                            As to the full range of practical statistics, that comes under Normal Distribution, NOT STANDARD ND. STANDARD is limited to a mean of zero and SD of 1.
                            The calculation I use (Black Sholes formula) is based on Standard ND, NOT ND.
                            With ND, the user's number will matter since the mean is different, e.g. if the user enters 1,000 and the mean is 900 the outcome is not the same as if the user enters 950. But in Standard ND, the outcome will not change (only by an infinitesimal decimal) if the user's number is >.68 because the mean is zero. When I use 100%, its for approximation purpose and the infinitesimal difference has to do with the number of iteration to reach convergence.

                            Comment


                              #44
                              Re: Cumulative Standard Normal Distribution

                              Originally posted by G Gabriel View Post
                              ... no matter what number the user enters, if it is more than .68 (One Standard Deviation of zero) the outcome will be the same because ...
                              You can think that if you wish, but you are wrong. Yes, the cumulative distribution measures the area under the bell curve and that bell curve goes on forever ... it doesn't stop at 1SD.

                              If you want to develop a function that gives the same results for all entries above 1 SD, go ahead. It will be about as useful in statistical applications as an automobile with two wheels, a calculator whose numerical keypad only has six buttons for numbers, or a payroll application which can only handle salaries up tyo five digits.

                              Comment


                                #45
                                Re: Cumulative Standard Normal Distribution

                                Steve,

                                This thread reminds me of why I had such a difficult time w. statistics in college.
                                Peter
                                AlphaBase Solutions, LLC

                                [email protected]
                                https://www.alphabasesolutions.com


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