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→Continuous probability distributions: Replace "less than" by "less than or equal to" in the section "Continuous probability distributions". |
→Continuous probability distributions: Added the reason why the quantile function is the inverse of the corresponding cumulative distribution function if it is monotonically increasing. |
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<math>F_X(x) = P(X \leq x)</math>.<ref name=":0" />
The [[Cumulative distribution function|CDF]] gives the probability that the random variable <math>X</math> is less than or equal to the value <math>x</math>. Therefore, the first quartile is the value of <math>x</math> when <math>F_X(x) = 0.25</math>, the second quartile is <math>x</math> when <math>F_X(x) = 0.5</math>, and the third quartile is <math>x</math> when <math>F_X(x) = 0.75</math>.<ref>{{Cite web|url=https://math.bme.hu/~nandori/Virtual_lab/stat/dist/CDF.pdf|title=6. Distribution and Quantile Functions|website=math.bme.hu}}</ref> The values of <math>x</math> can be found with the [[quantile function]] <math>Q(p)</math> where <math>p = 0.25</math> for the first quartile, <math>p = 0.5</math> for the second quartile, and <math>p = 0.75</math> for the third quartile. The quantile function is the inverse of the cumulative distribution function if the cumulative distribution function is [[Monotonic function|monotonically increasing]] because the [[Bijection|one-to-one correspondence]] between the input and output of the cumulative distribution function holds.
== Outliers ==
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