5 The cumulative distribution function

We know that the normal probability density function f ( x ) is given by the formula

f ( x ) = 1 σ 2 π e ( x μ ) 2 2 σ 2

and so the cumulative distribution function F ( x ) is given by the formula

F ( x ) = 1 σ 2 π x e ( u μ ) 2 2 σ 2 d u

In the case of the cumulative distribution for the standard normal curve, we use the special notation Φ ( z ) and, substituting 0 and 1 for μ and σ 2 , we obtain

Φ ( z ) = 1 2 π z e u 2 2 d u

The shape of the curve is essentially ‘ S ’ -shaped as shown in Figure 9. Note that the curve runs from to + . As you can see, the curve approaches the value 1 asymptotically.

Figure 9

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Comparing the integrals

F ( x ) = 1 σ 2 π x e ( u μ ) 2 2 σ 2 d u and Φ ( z ) = 1 2 π z e v 2 2 d v

shows that

v = u μ σ and so d v = d u σ

and F ( x ) may be written as

F ( x ) = 1 σ 2 π ( x μ ) σ e v 2 2 σ d v

= 1 2 π ( x μ ) σ e v 2 2 d v = Φ ( x μ σ )

We already know, from the basic definition of a cumulative distribution function, that

P ( a < X < b ) = F ( b ) F ( a )

so that we may write the probability statement above in terms of Φ ( z ) as

P ( a < X < b ) = F ( b ) F ( a ) = Φ ( b μ σ ) Φ ( a μ σ ) .

The value of Φ ( z ) is measured from z = to any ordinate z = z 1 and represents the probability P ( Z < z 1 ) .

The values of Φ ( z ) start as shown below:

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 .5000 5040 5080 5120 5160 5199 5239 5279 5319 5359
0.1 .5398 5438 5478 5517 5577 5596 5636 5675 5714 5753
0.2 .5793 5832 5871 5909 5948 5987 6026 6064 6103 6141

You should compare the values given here with the values given for the normal probability integral (Table 1 at the end of the Workbook). Simply adding 0.5 to the values in the latter table gives the values of Φ ( z ) . You should also note that the diagrams shown at the top of each set of tabulated values tells you whether you are looking at the values of Φ ( z ) or the values of the normal probability integral.

Exercises
  1. If a random variable X has a standard normal distribution find the probability that it assumes a value:
    1. less than 2.00
    2. greater than 2.58
    3. between 0 and 1.00
    4. between 1.65 and 0.84
  2. If X has a standard normal distribution find k in each of the following cases:
    1. P ( X < k ) = 0.4
    2. P ( X < k ) = 0.95
    3. P ( 0 < X < k ) = 0.1
    1. 0.9772
    2. 0.0049
    3. 0.3413
    4. 0.1510
    1. 0.2533
    2. 1.6450
    3. 0.2533