Symbols
Sample space
Event space
The probability of the event \(A\)
The probability that the event \(A\) occurs given that event \(B\) occurs.
A probability mass function (pmf)
The expected value of the random variable \(X.\)
The standard deviation of \(X.\)
The variance of \(X.\)
The conditional pdf of \(X\) given
The covariance between \(X\) and \(Y.\)
The correlation between \(X\) and \(Y.\)
The expected value of \(X\) given \(Y.\)
Key Words
The identity \(P(A|B) = P(B|A)P(A)/P(B).\) See the lesson for the more general version.
A random variable that has probability \(0\) of being any particular value, but has positive probability of taking values in open intervals \((a, b).\)
The correlation between random variables \(X\) and \(Y\) is a measure of the change in one of the random variables with respect to the other that is scaled to be between \(-1\) and \(1.\)
The covariance between random variables \(X\) and \(Y\) is a measure of the change in one of the random variables with respect to the other.
The cumulative distribution function of a random variable \(X\) is the function \(F(t) = P(X \leq t).\)
A random variable \(X\) is discrete if there exists a discrete set \(D\) such that \(P(X \in D) = 1.\)
The mean, or average value.
The set of all measurable events.
Events \(A\) and \(B\) are independent if \(P(A \cap B) = P(A)P(B).\)
Random variables \(X\) and \(Y\) are independent if the events \(\{X \in A\}\) and \(\{Y \in B\}\) are independent for all (measurable) subsets \(A, B\) of \(\mathbb{R}.\)
The median of a random variable \(X\) is the value \(c\) such that \(P(X \leq c) \geq \frac{1}{2}\) and \(P(X \geq c) \geq \frac{1}{2}.\)
The \(n\)th moment of a random variable \(X\) is \(E[X^n].\)
The moment generating function of a random variable \(X\) is the function \(M(s) = E[e^{sX}].\)
The \(p\)th percentile of a random variable \(X\) is the value \(c_p\) such that \(P(X \leq c_p) \geq p\) and \(P(X \geq c_p) \geq 1-p.\)
A function \(P:\mathcal{E} \rightarrow[0,1]\) such that \(P(\Omega)=1\) and if \(A_1, A_2, \dots\) are disjoint, \(P\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty P(A_i).\)
The probability density function, \(f,\) of a continuous random variable \(X,\) is the function such that \(P(X \in (a, b)) = \int_a^b f(x)dx\) for any open interval \((a,b).\)
The probability mass function, \(p,\) of a discrete random variable \(X\) is the function defined by \(p(x) = P(X = x).\)
The first quartile of a random variable is the \(25\)th percentile. The second quartile is the \(50\)th percentile, also known as the median. The third quartile is the \(75\)th percentile.
A function \(X:\Omega\rightarrow\mathbb{R}.\)
The set of all outcomes of an experiment.
A mean value of the distance of a random variable \(X\) to its mean. The square root of the variance.
A measure of the spread of values of a random variable. The square of the standard deviation.