Chapter 3 Flashcards
What is an experiment?
An experiment is a set of rules that governs a specific procedure, which can be indefinitely repeated and has a well-defined set of outcomes. An experiment with only one possible outcome is called a deterministic experiment, while an experiment with two or more possible outcomes is called a random experiment
What is a trial?
A trial is any performance or exercise of a defined experiment.
What is an outcome?
Result of a given trial.
What is the sample space?
A set \omega of all possible outcomes of an experiment
What does the classical definition of probability assume?
A discrete and uniform distribution of probabilities.
What is a probability space?
Mathematical model of real-world processes.
Consists of:
- Sample space
- Set of Events
- Probabilities
What are Kolmogorovβs Axioms?
- The probability of an event \omega is non-negative
- The probability that some event in the entire sample space will occur is 1
- Sum of probabilities is homogeneous
What is a stochastic variable?
A variable representing the outcome of a naturally real-valued random experiment
or, a a function X mapping the probability space of a random experiment to real numbers
What is the stochastic process?
Stochastic processes are used to describe probabilities of non-deterministic systems.
What is Ergodicity?
If all the statistics of a process ππ may be determined from a single function ππ π0 of the process, it is said to be ergodic β that is, it behaves the same whether analyzed over time or averaged over space.
What is the probability distribution function ?
The probability distribution function (pdf) (probability mass, probability density) describes the probability of a stochastic variable taking certain values.
What is the probability mass function?
Probability distribution function for discrete probability distribution
What is Probability density function?
Probability distribution function for Continuous probability distribution
What is the cumulative distribution function?
The cumulative distribution function (cdf) describes the probability that a stochastic variable π with a given probability distribution (discrete or continuous) will be found at a value less than or equal to π₯.
What are Statistical Moments?
A moment can be seen as a quantitative measure of the shape of a set of points, i.e. the moments describe how a probability distribution is shaped
What are the formulas for the k-th moment of a random variable X?
Continuous:
E[X^k] = \int_{-\infty}^{\infty} x^k p(x) dx
Discrete:
E[X^k] = \sum_{i=1}^N x_i^k p(x_i)
What are the formulas for the k-th central moment of a random variable X?
Continuous:
E[(X-\mu)^k] = \int_{-\infty}^{\infty} (x-\mu)^k p(x) dx
Discrete:
E[(X-\mu)^k] = \sum_{i=1}^N (x_i-\mu)^k p(x_i)
What are the formulas for the k-th standardized moment of a random variable X?
Continuous:
E[((X-\mu)/\sigma)^k] = (1/\sigma)^k * \int_{-\infty}^{\infty} (x-\mu)^k p(x) dx
Discrete:
E[((X-\mu)/\sigma)^k] = (1/\sigma)^k * \sum_{i=1}^N (x_i-\mu)^k p(x_i)
What is expected value / expectation / mean? What are its formulas
1st moment of x
Discrete:
E[x] = \sum_{i=1}^N x_i p_d(x_i) = \micro
Continuous E[x] = \int_{-\infty}^{\infty} xp(x)dx = \micrp
What is variance \sigma? What are its formulas?
Second statistical moment. Describes how far away the number lie from the mean.
Discrete: Var[x] = \sum_{i=1}^N (x_i - \micro)^2 * p_d(x_i) = E[(X-E[X])^2]
Continuous:
Var[x] = \int_{-\infty}^{\infty} (x-\micro)^2 p(x) dx = E[(X-E[X])^2]
What is Skewdness? What are its formulas?
Normalized third central moment. Indicator of asymmetry
Discrete: S[x] = (1/\sigma^3) * \sum_{i=1}^N (x_i - \micro)^3 * p_d(x_i)
Continuous:
S[x] = (1/\sigma^3) * \int_{-\infty}^{\infty} (x-\micro)^3 p(x) dx
What is Kurtosis. What are its formulas?
Fourth normalized moment. Degree of peakedness
Discrete: K[x] = (1/\sigma^4) * \sum_{i=1}^N (x_i - \micro)^4 * p_d(x_i)
Continuous:
K[x] = (1/\sigma^4) * \int_{-\infty}^{\infty} (x-\micro)^4 p(x) dx
What is covariance?
The covariance (or cross covariance) of two stochastic variables π and π is a measure of how the variables change together.
What is the formula for the correlation coefficient?
\rho = \frac{Cov(x,y)}{\sigma_x \sigma_y}
What is the formula for covariance?
Cov(X,Y) = E[(X - E[X])(Y-E[Y])]
What is conditional probability?
The conditional probability is the probability of an event under certain circumstances, more precisely when the sample space is limited to another event.
What is the formula for conditional probability?
Let π΄ and π΅ be two events. Then the conditional probability of π΅ given that π΄ has occurred is given by :
P(B|A) = \frac{P(A N B)}{P(A)}
What is Bayesβ Theorem?
P(B|A) = \frac{P(B)P(A|B)}{P(A)}
What is the Law of Large Number?
For π samples, the probability of one sample can be approximated by the empirical probability π π β 1/π
Sample mean will approach actual mean as N approaches infinity
What is the central limit theorem?
The Central Limit Theorem states, that a scaled version of the sample average is normally distributed about the true mean
What are some additional distributions?
- X^2 distributions
- F distributions
- t- statistic / student-t distributions
What is a type 2 error?
The sample shows correct when the statement is false
What is type 1 Error?
The sample shows incorrect, when correct
What is the hypothesis testing process?
- Formulate hypothesis
- Formally state hypothesis
- Decide which relevant test-statistic, i.e. which metric π shall be used in the decision-making process
- Determine the distribution of the test-statistic, under the null hypothesis
- Choose a level of significance πΌ.
- Under the null-hypothesis, with the distribution of the test-statistic, a threshold value for the data can be determined.
- The threshold defines a βcritical regionβ for the possible outcomes of the experiment, where π»_0 is rejected
- Compute the observed value of the test-statistic t
- If π‘πππ is in the critical region, π»0 can be rejected and π»1 accepted.
What is E[X]?
It is the expectation / first statistical moment
How are the expectation and the variance of the variance of the x^2 distribution defined?
E[Z] = N
Var[z] = 2N
where N is the degrees of freedom
How are the expectation and the variance of the variance of the t-statistic distribution defined?
E[T] = 0
Var[T] = 2/(N-2)
where N is the degrees of freedom
