Flashcards
Statistical Inference
Cards in this deck (18)
- 1Normal-chi squared Relationship (p.53)Let \(X\) have the standard normal distribution. Then the pdf of \( Y = X^2 \) is chi squared with 1 degree of freedom.
- 2Moments and Central Moments (p.59)For each integer \( n \), the \(n\)th moment of \( X \) (or \(F_X(x) \) ), \( \mu_n^{'} \), is \[ \mu_n^{'} = \text{E} X^n \]
The \(n \)th central moment of \( X \), \( \mu_n \), is
\[ \mu_n = \text{E}(X- \mu)^n \]
where \( \mu = \mu_1^{'} = \text{E}X \) - 3Location family (p.116)Let \( f(x) \) be any pdf. Then the family of pdfs \( f(x - \mu) \), indexed by the parameter \( \mu \), \( -\infty < \mu < \infty \), is call the location family with standard pdf \( f(x) \) and \( \mu \) is called the location parameter for the family.
- 4Scale Family (p. 119)Left \( f(x) \) be any pdf. Then for any \( \sigma > 0 \), the family of pdfs \( (1/ \sigma)f(x/ \sigma) \), indexed by the parameter \( \sigma \), is called the scale family with standard pdf \( f(x) \) and \( \sigma \) is called the scale parameter of the family.
- 5Location-Scale Family (p.119)Left \( f(x) \) be any pdf. Then for any \( \mu \), \( -\infty < \mu < \infty \), and any \( \sigma > 0 \), the family of pdfs \( (1/ \sigma) f((x- \mu)/ \sigma) \), indexed by the parameter \( (\mu, \sigma) \), is called the location-scale family with standard pdf \( f(x) \); \( \mu \) is called the location parameter and \( \sigma \) is called the scale parameter.
- 6Union (p.2)\[ A \cup B = \{x: x \in A \text{ or } x \in B\} \]
- 7Intersection p.2\[ A \cap B = \{ x: x \in A \text{ and } x \in B \} \]
- 8Disjoint and Pairwise Disjoint p.5Two Events A and B are disjoint (or mutually exclusive) if \( A \cap B = \emptyset \). The events \( A_1, A_2, \dotsc \) are pairwise disjoint (or mutual exclusive) if \( A_i \cap \ A_j = \emptyset \) for all \( i \neq j \).
- 9Complement p.3The complement of \( A \), written \( A^c \), is the set of all elements that are not in \( A \)
\[ A^c = \{ x:x \notin A \} \] - 10Commutativity p.3For any three events, \( A\), \( B\), and \( C \), defined on a sample space \( S \),
\[A \cup B = B \cup A \]
\[ A \cap B = B \cap A \] - 11Associativity p. 5For any three events, \( A\), \( B\), and \( C \), defined on a sample space \( S \),
\[ A \cup (B \cup C ) = (A \cup B) \cup C \]
\[ A \cap (B \cap C) = (A \cap B) \cap C \] - 12Cumulative Distribution Function (CDF) (Def. 1.5.1)The cumulative distribution function or cdf of a random variable \( X \) , denoted by \( F_X(x) \), is defined by:
\[ F_X(x) = P_X(X \leq x) , \text{for all } x \] - 13Conditions for a CDF, Theorem 1.5.3. The function F(x) is a cdf if and only if the following three conditions hold:1. \(\lim_{x \to -\infty} F(x) = 0 \) and \( \lim_{x \to \infty} F(x)=1 \)
2. \( F(x) \) is a nondecreasing function of x
3. \( F(x) \) is right continous; that is, for every number \( x_0 \) , \( \lim_{x \downarrow x_0} F(x) = F(x_0) \) - 14Random Variable (Def. 1.4.1)A random variable is a function from a sample space \( S \) into the real numbers. (or the outcome of an experiment -Ash)
- 15Mutual independence of a collection of events (Def. 1.3.12)A collection of events \( A_1, \dotsc ,A_n \) are mutually independent if for any sub collection \( A_{i_1}, \dotsc ,A_{i_k} \) , we have \( P\left( \displaystyle \bigcap_{j=1}^k A_{i_j} \right) = \displaystyle \prod_{j=1}^k P(A_{i_j}) \)
- 16Sample Space (Def. 1.1.1)The set, \( S \), of all possible outcomes of a particular experiment
- 17EventAny collection of possible outcomes of an experiment, that is, any subset of \( S \) (including \( S \) itself).
- 18Continuous and Discrete RVs (Def. 1.5.7)A random variable \( X \) is continuous if \( F_X(x) \) is a continuous function of \( x \). A random variable \( X \) is discrete is \( F_X(x) \) is a step function of \( x \).
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- None Stats Flashcards on Statistical Inference, created by Jared Carlson on 05/03/2015.
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- Jared Carlson
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