For every real number x, the cumulative distribution function of a real-valued random variable X is given by

For each puzzler, one correct answer is chosen at random, with the winner receiving a $ 26 gift certificate to the Car Talk store, referred to as the " Shameless Commerce Division ".

For certain types of point sets, such as a uniform random distribution, by intelligently picking the splitting lines the expected time can be reduced to O ( n log log n ) while still maintaining worst-case performance.

For the quantitative analysis, ten articles were selected at random – circumcision, Charles Drew, Galileo, Philip Glass, heart disease, IQ, panda bear, sexual harassment, Shroud of Turin and Uzbekistan – and letter grades of A – D or F were awarded in four categories: coverage, accuracy, clarity, and recency.

For elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible.

For example, consider a model which gives the probability density function of observable random variable X as a function of a parameter θ.

For example, children should hold a greater love for their parents than for random strangers.

For example, it is common for digital balances to exhibit random error in their least significant digit.

For example, seven shuffles of a new deck leaves an 81 % probability of winning New Age Solitaire where the probability is 50 % with a uniform random deck ( Mann, especially section 10 ).

For example, one can assign a random number to each card, and then sort the cards in order of their random numbers.

Informally speaking, the prime number theorem states that if a random integer is selected in the range of zero to some large integer N, the probability that the selected integer is prime is about 1 / ln ( N ), where ln ( N ) is the natural logarithm of N. For example, among the positive integers up to and including N = 10 < sup > 3 </ sup > about one in seven numbers is prime, whereas up to and including N = 10 < sup > 10 </ sup > about one in 23 numbers is prime ( where ln ( 10 < sup > 3 </ sup >)= 6. 90775528. and ln ( 10 < sup > 10 </ sup >)= 23. 0258509 ).

For example, the notion of gauge invariance forms the basis of the well-known Mattis spin glasses, which are systems with the usual spin degrees of freedom for i = 1 ,..., N, with the special fixed " random " couplings Here the ε < sub > i </ sub > and ε < sub > k </ sub > quantities can independently and " randomly " take the values ± 1, which corresponds to a most-simple gauge transformation This means that thermodynamic expectation values of measurable quantities, e. g. of the energy are invariant.

#* For security purposes, the integers p and q should be chosen at random, and should be of similar bit-length.

For a continuous random variable, the probability of any specific value is zero, whereas the probability of some infinite set of values ( such as an interval of non-zero length ) may be positive.

For the most part, statistical inference makes propositions about populations, using data drawn from the population of interest via some form of random sampling.

For example, one may assume that the distribution of population values is truly Normal, with unknown mean and variance, and that datasets are generated by ' simple ' random sampling.

For example, every continuous probability distribution has a median, which may be estimated using the sample median or the Hodges-Lehmann-Sen estimator, which has good properties when the data arise from simple random sampling.

For n independent and identically distributed continuous random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >, ..., X < sub > n </ sub > with cumulative distribution function G ( x ) and probability density function g ( x ) the range of the X < sub > i </ sub > is the range of a sample of size n from a population with distribution function G ( x ).

For n nonidentically distributed independent continuous random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >, ..., X < sub > n </ sub > with cumulative distribution functions G < sub > 1 </ sub >( x ), G < sub > 2 </ sub >( x ), ..., G < sub > n </ sub >( x ) and probability density functions g < sub > 1 </ sub >( x ), g < sub > 2 </ sub >( x ), ..., g < sub > n </ sub >( x ), the range has cumulative distribution function

For n independent and identically distributed discrete random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >, ..., X < sub > n </ sub > with cumulative distribution function G ( x ) and probability mass function g ( x ) the range of the X < sub > i </ sub > is the range of a sample of size n from a population with distribution function G ( x ).

For the univariate case, Indeed, for random variables X and Y, the expectation of their product is an inner product.

For every finite subset, the k-tuple is a random variable taking values in.

For example, a pianist might simply sit and start playing chords, melodies, or random notes that come to mind in order to find some inspiration, then build on the discovered lines to add depth.

For the low-temperature measurements the sample was cooled by a cold nitrogen gas flow method similar to that of Andrew and Eades.

For the albumin method, equal volumes of 30% bovine albumin, sample and 2% cells suspended in saline were allowed to stand at room temperature for 1 hr and then were centrifuged at 1000 rpm for 1 Aj.

For purposes of sample selection only ( individual tests were given later ) we obtained group test scores of reading achievement and intelligence from school records of the entire third-grade population in each school system.

The Wechsler Intelligence Scale For Children was administered to each sample third-grade child by a clinical worker.

For example, on the basis of the regression equation, a child with an I.Q. of 120 in this sample would be expected to earn an achievement test score of 4.8 ( grade equivalent ).

For example, consider the sample space

For example, if n samples are taken from a Cauchy distribution, one may calculate the sample mean as:

Other, more precise and robust methods have been developed For example, the truncated mean of the middle 24 % of the sample order statistics produces an estimate for x < sub > 0 </ sub > that is more efficient than using either the sample median or the full sample mean.

For example in a paper reporting on a study involving human subjects, there typically appears a table giving the overall sample size, sample sizes in important subgroups ( e. g., for each treatment or exposure group ), and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, and the proportion of subjects with related comorbidities.

For modelling terrain or other objects given a set of sample points, the Delaunay triangulation gives a nice set of triangles to use as polygons in the model.

For a given sample, the " error " of the estimator is defined as

For a given sample, the sampling deviation of the estimator is defined as

For the sample image above the reversible mapping between 9-bit codes and bytes is shown below.

For the sample 3x5 image above, the following 9-bit codes represent " clear " ( 100 ) followed by image pixels in scan order and " stop " ( 101 ).

For example, a survey of high school students to measure teenage use of illegal drugs will be a biased sample because it does not include home-schooled students or dropouts.

For a typical liquid crystal sample, S is on the order of 0. 3 to 0. 8, and generally decreases as the temperature is raised.

For example, oxygen makes up about < sup > 8 </ sup >/< sub > 9 </ sub > of the mass of any sample of pure water, while hydrogen makes up the remaining < sup > 1 </ sup >/< sub > 9 </ sub > of the mass.

# For progesterone testing, a veterinarian can test a blood sample for progesterone.

For example, a food sample might be analyzed on three different nutrient mediums designed to indicate the presence of " total " bacteria ( conditions where many, but not all, bacteria grow ), molds ( conditions where the growth of bacteria is prevented by, e. g., antibiotics ) and coliform bacteria ( these indicate a sewage contamination ).

For the NAA procedure to be successful the specimen or sample must be selected carefully.

For example, if the experiment is tossing a coin, the sample space is the set