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Normal Distribution
Normal Distribution
• Turning from discrete to continuous distributions, in this section
we discuss the normal distribution.
• This is the most important continuous distribution because in
applications many random variables are normal random variables
(that is they have a normal distribution) or they are approximately
normal or can be transformed into normal random variables in a
relatively simple fashion.
• Furthermore, the normal distribution is a useful approximation of
more complicated distributions and it also occurs in the proofs of
various statistical tests.
Normal Distribution
• The normal distribution or Gauss Distribution is defined as the
distribution with the density
• Where exp is the exponentiel function with base 𝑒 = 2.718…
Normal Distribution
• This is simpler than it may at first look. F(x) has these features:
• 1. 𝜇 is the mean and 𝜎 the standart deviation.
• 2. 1/(𝜎 2𝜋) is a constant factor that makes the area under the
curve of f(x) from -∞ to ∞ equal to 1.
• 3. The curve of f(x) is symmetric with respect to x = 𝜇 because the
exponent is quadratic. Hence for 𝜇 = 0 it is symmetric with
respect to the y-axis x=0 (bell-shaped curves)
• 4. The exponential function in (1) goes to zero very fast – the
faster the smaller the standart deviation 𝜎 is.
Normal Distribution
Distribution Function F(x)
• The normal distribution has the distribution function
Distribution Function F(x)
• Here we needed x as the upper limit of integration and wrote 𝑣
(instead of x) in the integrand.
• For the corresponding standardized normal distribution with mean
0 and standart deviation 1 we denote F(x) by 𝛷(z).
• Then we simply have from (2)
Distribution Function F(x)
• This integrand can not be integrated by one of the methods of
calculus. But this is no seriou handicap because its values can be
optained from a Table***.
• These vaues are needed in working with the normal distribution.
• The curve of 𝛷(z) is S-shaped.
• It increases monotone from 0 to 1 and intersects the vertical axis
at ½.
Reltion beteen F(x) and 𝛷(z)
• Although your table will give you the values of F(x) in (2) with any
𝜇 and 𝜎 directly, it is important to comprehand that and such an
F(x) can be expressed in terms of the tabulated standart 𝛷(z), as
follows
Use of normal table
• The distribution function F(x) of the normal distribution with any
𝜇 and 𝜎 is related to the standardized distribution function 𝛷(z) in
(3) by the formula
Proof
• Comparing (2) and (3) we see that we should set
Then 𝑣 gives x
as the upper limit of inegration.
Proof
• Also 𝑣 - 𝜇 = 𝜎𝑢, thus d𝑣 = 𝜎 𝑑𝑢.
• Together, since 𝜎 drops out,
• Probabilities corresponding to intervals will be needed quite
frequently in statistics.
Normal Probabilities for Intervals
• The probability that a normal random variable X with mean 𝜇 and
standart deviation 𝜎 assume any value in an interval a < x ≦ b is
Numeric Values
• In practical work with the normal distribution is good to remember
about 2/3 of all values of X to be observed will lie between 𝜇 ∓ 𝜎,
about 95% between 𝜇 ∓ 2𝜎, and practivally all between the three
sigma limits 𝜇 ∓ 3𝜎. More precisely,
Numeric Values
• The formulas in (6) Show that a value deviating from 𝜇 by more
than 𝜎, 2𝜎 or 3𝜎 will occur in one of about 3,20 and 300 trials
respectively.
Numeric Values
• In statistic tests we shall ask, conversly, for the intervals that
correspond to certain given probabilities; practically most
important are the probabilities of 95%, 99% and 99.9%.
• For these tbale gives the answers 𝜇 ∓ 2𝜎, 𝜇 ∓ 2.6𝜎 and 𝜇 ∓ 3.3𝜎,
respectively.
Working with the Normal Tables
• There are two normal tables A7 and A8.
• If you want probabilities, use A7.
• If probabilities are given and corresponding intervals or x-values
are wanted, use A8.
• The following examples are typical.
Example 1 (Reading Entries form A7)
• If X is standardized normal (so that 𝜇=0 and 𝜎=1), then
Example 2
(Probabilities for Given Intervals,A7)
• Let X be normal with mean 0.8 and variance 4(so that 𝜎=2). Then
by (4) and (5)
• Or if you like it better(similarly in the other case)
Example 3
(Unknown Values c for Given Probabilties, A8)
• Let X be normal with mean 5 and variance 0.04. (hence standart
deviation 0.2). Find c or k corresponding to the given probability.
Example 4 (Defectives)
• In a production iron rods let the diameter X be normally
distributed wit mean 2 in. And standart deviation 0.008 in.
• (a) Wht percantage of defectives can we expert if we set the
tolarance limits at 2 ± 0.02 in?
• (b) How should we set the tolerance limits to allow for 4%
defectives?
Example 4 (Defectives) Solution
1
1
4
• (a)
% because from (5) and A7we obtain for the complementary
event the probability
Example 4 (Defectives) Solution
• (b) 2 ± 0.0164 because, for the complementary event, we have
Or
So that A8 gives
Normal Approximation
of the Binomial Distribution
• The probability function of the binomial distribution is
• If n is large, the binomial coefficents and power become very
inconvenient.
• It is of great practical (ad therotical) importance that, the normal
distribution proveides a good approximation of the binomial
distribution, according to the following theorem, one of the most
important theorems in al probability theory.
Limit Theorem of De Moivre and Laplace
• For large n,
• Here f is given by (8). The function
Limit Theorem of De Moivre and Laplace
• Is the density of the normal distribution with mean 𝜇 = np and the
variance 𝜎 2 = npq (the mean and the variance of the binomial
distributon).
• The symbol ~ (read assimptotically equal) means that the ratio of
both sides approaches 1 as n approaches ∞. Furthermore, for ny
nonnegativ integer 𝑎 and b ( > 𝑎),