Probability: Difference between revisions

Probability (view source)

985 bytes added , 4 February 2023

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 * For all events <math>A</math> and <math>B</math>, <math>A \subset B \implies P(A) \leq P(B)</math>
 {{hidden | Proof | }}
+===Conditional Probability===
+<math>P(A|B)</math> is the probability of event A given event B.<br>
+Mathematically, this is defined as <math>P(A|B) = P(A,B) / P(B)</math>.<br>
+Note that this can also be written as <math>P(A|B)P(B) = P(A, B)</math>
+With some additional substitution, we get '''Baye's Theorem''':
+<math>
+P(A|B) = \frac{P(B|A)P(A)}{P(B)}
+</math>
+==Random Variables==
+A random variable is a variable which takes on a distribution rather than a value.
 ===PMF, PDF, CDF===
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 The comulative distribution function (CDF) is <math>F(x) = P(X \leq x)</math>.<br>
 The CDF is the prefix sum of the PMF or the integral of the PDF. Likewise, the PDF is the derivative of the CDF.
+===Joint Random Variables===
+Two random variables are independant if <math>f_{X,Y}(x,y) = f_X(x) f_Y(y)</math>.
+Otherwise, the marginal distribution is <math>f_X(x) = \int f_{X,Y}(x,y) dy</math>.
+===Conditional Random Variables===
+...
+===Change of variables===
+Let <math>g</math> be a monotonic increasing function and <math>Y = g(X)</math>.
+Then <math>F_Y(y) = P(Y \leq y) = P(X \leq g^{-1}(y)) = F_X(g^{-1}(y))</math>.
+And <math>f_Y(y) = (d/dy)F_Y(y) = (d/dy) F_X(g^{-1}(y)) = f_X(g^{-1}(y)) (d/dy)g^{-1}(y)</math>
 ==Expectation and Variance==