sample space

Examples of sample space in the following topics:

• The Multiplication Rule

  • If A and B are two events defined on a sample space, then: P(A AND B) = P(B) · P(A|B).

• Terminology

  • A sample space is a set of all possible outcomes.
  • The uppercase letter S is used to denote the sample space.
  • The sample space has four outcomes.
  • A conditional reduces the sample space.
  • The sample space S = { 1, 2, 3, 4, 5, 6 } .

• Lab 2: Central Limit Theorem (Cookie Recipes)

  • Record your samples below.
  • Record them in the spaces provided.
  • Thistime, make the samples of size n = 10.
  • Record the samples below in Table 7.6.
  • Record them in the spaces provided.

• Estimation

  • Out of a random sample of 200 people, 106 say they support the proposition.
  • Thus in the sample, 0.53 ($\frac{106}{200}$) of the people supported the proposition.
  • It is rare that the actual population parameter would equal the sample statistic.
  • Bias leads to a sample mean that is either lower or higher than the true mean .
  • An estimate of expected error in the sample mean of variable $A$, sampled at $N$ locations in a parameter space $x$, can be expressed in terms of sample bias coefficient $\rho$ -- defined as the average auto-correlation coefficient over all sample point pairs.

• Quality of Sound

  • As samples are placed closer together in time, higher frequencies can be reproduced.
  • According to the sampling theorem, any signal with bandwidth B can be perfectly described by more than 2B samples per second.
  • Audio must be sampled at above 40kHz: 44.1kHz for CD recordings and 48kHz for DVD recordings.
  • The amount of space required to store PCM depends on the number of bits per sample, the number of samples per second, and the number of channels.
  • For CD audio, this is 44,100 samples per second, 16 bits per sample, and 2 channels for stereo audio leading to 1,411,200 bits per second.

• Standard Error

  • However, different samples drawn from that same population would in general have different values of the sample mean.
  • The standard error of the mean (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all possible samples (of a given size) drawn from the population.
  • SEM is usually estimated by the sample estimate of the population standard deviation (sample standard deviation) divided by the square root of the sample size (assuming statistical independence of the values in the sample):
  • $s$ is the sample standard deviation (i.e., the sample-based estimate of the standard deviation of the population), and
  • If values of the measured quantity $A$ are not statistically independent but have been obtained from known locations in parameter space $x$, an unbiased estimate of the true standard error of the mean may be obtained by multiplying the calculated standard error of the sample by the factor $f$:

• Mann-Whitney U-Test

  • For small samples a direct method is recommended.
  • Call this "sample 1," and call the other sample "sample 2. "
  • where $n_1$ is the sample size for sample 1, and $R_1$ is the sum of the ranks in sample 1.
  • Note that it doesn't matter which of the two samples is considered sample 1.
  • $U$ remains the logical choice when the data are ordinal but not interval scaled, so that the spacing between adjacent values cannot be assumed to be constant.

• Linear Vector Spaces

  • If we record one seismogram one second in length at a sample rate of 1000 samples per second, then we can put these 1000 bytes of information in a 1000-tuple
  • where $s_i$ is the i-th sample, and treat it just as we would a 3-component physical vector.
  • Whereas our use of vector spaces is purely abstract.
  • In the case of $n=1$ the vector space $V$ and the field $F$$F$ are the same.
  • So trivially, $F$ is a vector space over $F$ .

• Introduction to The Sampling Theorem

  • $f_s$ is called the sampling frequency.
  • Putting all this together, one can show that the band limited function $f(t)$ is completely specified by its values at the countable set of points spaced $1/2f_s$ apart:
  • So the sampling theorem says take the value of the function, sampled every $1/2f_s$ , multiply it by a sinc function centered on that point, and then sum these up for all the samples.
  • Suppose we sample this signal at a sampling period of $T_s$ .
  • In a 1928 paper Nyquist laid the foundations for the sampling of continuous signals and set forth the sampling theorem.

• Interpreting Phase Diagrams

  • Phase diagrams are divided into three single phase regions that cover the pressure-temperature space over which the matter being evaluated exists: liquid, gaseous, and solid states.
  • By focusing attention on distinct single phase regions, phase diagrams help us to understand the range over which a particular pure sample of matter exists as a particular phase.
  • Phase diagrams can also be used to explain the behavior of a pure sample of matter at the critical point.
  • The critical point, which occurs at critical pressure (Pcr) and critical temperature (Tcr), is a feature that indicates the point in thermodynamic parameter space at which the liquid and gaseous states of the substance being evaluated are indistinguishable.
  • At temperatures above the critical temperature, the kinetic energy of the molecules is high enough so that even at high pressures the sample cannot condense into the liquid phase.