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The contaminated normal distribution is a special form of mixed distribution . It plays a major role in robustness studies of estimators and tests .
The real random variable has a contaminated normal distribution if its density function is in the form
![X \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/7028e89b7722d12ec0ea8780f26a9912456b63f5)
![{\ displaystyle f (x) = (1- \ varepsilon) {\ frac {1} {{\ sqrt {2 \ pi}} \ sigma _ {1}}} \ operatorname {e} ^ {- {\ frac { 1} {2}} \ left ({\ frac {x- \ mu _ {1}} {\ sigma _ {1}}} \ right) ^ {2}} + \ varepsilon {\ frac {1} {{ \ sqrt {2 \ pi}} \ sigma _ {2}}} \ operatorname {e} ^ {- {\ frac {1} {2}} \ left ({\ frac {x- \ mu _ {2}} {\ sigma _ {2}}} \ right) ^ {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34cb79e9eb082827ac1ace59fd4a10cfd4f99ab5)
with , i.e. as a convex combination of two normal distribution density functions.
![{\ displaystyle \ varepsilon \ in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d472b1b7207f57be05aaefccd6651d486dd59a8c)
The distribution function then has the form
-
.
The distribution function of a normally distributed random variable applies here .
![{\ displaystyle \ mu _ {1}, \ mu _ {2} \ in \ mathbb {R}, \ sigma _ {1}, \ sigma _ {2}> 0, \ operatorname {N} (\ mu, \ sigma ^ {2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e581a624777aed52090ec0c29ede8b380506c3)
The following applies to the expected value and the variance :
,
.
Often, through additional conditions, such as special cases are derived ( scale-contaminated normal distribution ).
![{\ displaystyle \ mu _ {1} = \ mu _ {2} \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5bf5756d7bf88e4f4346cfebea702e9cd023214)
example
A manufacturer of electronic devices uses capacitors with a capacity of 5 nF, which it purchases from two manufacturers. Those manufactured by A show a slightly smaller scatter than those from B. 60% of the capacitors purchased come from manufacturer A, 40% from B. Assume that the capacitance of the capacitors from both manufacturers is normally distributed with parameters in a sufficiently wide range . Be and .
![{\ displaystyle \ mu _ {1}, \ mu _ {2}, \ sigma _ {1} ^ {2}, \ sigma _ {2} ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54fba2897f97536e345b5128cb9ee89ffcdf9ec1)
![{\ displaystyle \ mu _ {1} = \ mu _ {2} = 5 \, \ operatorname {nF} \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/655e5004ea377342a4f4795d224f9283d7f745bd)
![{\ displaystyle \ sigma _ {1} ^ {2} = 0 {,} 0144 \, \ operatorname {nF} ^ {2}, \ sigma _ {2} ^ {2} = 0 {,} 0225 \, \ operatorname {nF} ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c960fada0c7bec9e82d238a8a146acb7059146b)
A deviation of more than 10% from the nominal value of the capacitance is highly undesirable. What is the probability that a capacitor has a capacitance that differs by more than 10%?
A proportion of around 0.000361849 of all capacitors shows a deviation of more than 10% in terms of capacity.