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Suppose and random

Suppose that Y is the sum of n identically distributed independent random variables all with the same distribution as X.

Suppose that you are popping one hundred kernels of popcorn, and each kernel will pop at an independent, uniformly random time within the next hundred seconds.

Suppose we have a random sample of size n from a population,.

Suppose then that we have a random classifier that guesses that you have the disease with that same probability and guesses you do not with the same probability.

Suppose we have a set of observable random variables, with means.

Suppose for some unknown constants and unobserved random variables, where and, where < math > k , we have

Suppose the number of a man's sons to be a random variable distributed on the set

Suppose the random column vectors X, Y live in R n and R m respectively, and the vector ( X, Y ) in R n + m has a multivariate normal distribution whose variance is the symmetric positive-definite matrix

Suppose the probability distribution of a discrete random variable X puts equal weights on 1, 2, and 3:

Suppose at a future time a derivative ( e. g., a call option on a stock ) pays units, where is a random variable on the probability space describing the market.

Suppose the actual series of observed data that needs to be analysed has approximately the same variance properties as the random series used in the above empirical investigation.

Suppose contains independent random components, each of which has three possible realizations ( for example, future realizations of each random parameters are classified as low, medium and high ), then the total number of scenarios is.

Suppose further that we can generate a sample of replications of the random vector.

Suppose that we have a sample of realizations of the random vector.

Suppose U < sub > 1 </ sub >, ..., U < sub > n </ sub > are independent standard normally distributed random variables, and an identity of the form

Suppose that is a standard multivariate normal random vector ( here denotes the n-by-n identity matrix ), and if are all n-by-n symmetric matrices with.

Suppose X is a normally distributed random variable with known mean and unknown variance.

Suppose there is a sequence of random variables

Suppose a chord of the circle is chosen at random.

Suppose ( unrealistically ) that the number N is chosen by some random process that is independent of the batter's ability – say a coin is tossed after each at-bat and the result determines whether the scout will stay to watch the batter's next at-bat.

Suppose X is a normally distributed random variable with expectation μ and variance σ 2 .

Suppose that a random variable, X, is defined to be the time elapsed in a shop from 9 am on a certain day until the arrival of the first customer: thus X is the time a server waits for the first customer.

Suppose X is a discrete random variable whose values lie in the set

Suppose and variable

Suppose that ƒ is a function of more than one variable.

Suppose the system has one external variable x.

Suppose we consider the same example used in the ANOVA model with 1 qualitative variable: average annual salary of public school teachers in 3 geographical regions of Country A.

Suppose H is a non-decreasing function of a real variable.

Suppose that,, and are quantifier-free formulas and no two of these formulas share any free variable.

Suppose that a 2-satisfiability instance contains two clauses that both use the same variable x, but that x is negated in one clause and not in the other.

Suppose we want to add the two arrays x and y and also do something depending on the variable w. We have the following C code:

Suppose an experimenter performs 10 measurements all at exactly the same value of independent variable vector X ( which contains the independent variables X < sub > 1 </ sub >, X < sub > 2 </ sub >, and X < sub > 3 </ sub >).

Suppose we have a function f of a complex variable z which is not analytic, but happens to be differentiable with respect to its real and imaginary components separately.

Suppose that X is a random variable taking values in and let f be a uniquely decodable code from to where.

Suppose response variable Y is binary, that is it can have only two possible outcomes which we will denote as 1 and 0.

Suppose there exists an auxiliary random variable

Suppose for simplicity that a certain system is characterized by two variables-a dependent variable x and an independent variable t, where x is a function of t. Both x and t represent quantities with units.

Suppose the change in x is known under some new circumstance: to estimate the likely change in an outcome variable y, the gradient of the regression of y on x is needed, not y on w. This arises in epidemiology.

Suppose that a random variable J has a Poisson distribution with mean, and the conditional distribution of Z given is chi-squared with k + 2i degrees of freedom.

Suppose and X

) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub > 2 for all times i. Then the definition of the autocorrelation between times s and t is

Suppose that on these sets X and Y, there are two binary operations and that happen to constitute the groups ( X ,) and ( Y ,).

: Suppose X is a compact Hausdorff space and A is a subalgebra of C ( X, R ) which contains a non-zero constant function.

Suppose that X is a topological space.

Suppose that U: D → C is a functor from a category D to a category C, and let X be an object of C. Consider the following dual ( opposite ) notions:

Suppose ( A < sub > 1 </ sub >, φ < sub > 1 </ sub >) is an initial morphism from X < sub > 1 </ sub > to U and ( A < sub > 2 </ sub >, φ < sub > 2 </ sub >) is an initial morphism from X < sub > 2 </ sub > to U. By the initial property, given any morphism h: X < sub > 1 </ sub > → X < sub > 2 </ sub > there exists a unique morphism g: A < sub > 1 </ sub > → A < sub > 2 </ sub > such that the following diagram commutes:

An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well-order: Suppose X is a subset of R well-ordered by ≤.

Suppose that X is a regular space.

Suppose that X and Y are a pair of commuting vector fields.

Suppose we are given a topological space X.

Suppose that X and Y are two plane projective curves defined over a field F that do not have a common component ( this condition is true if both X and Y are defined by different irreducible polynomials, in particular, it holds for a pair of " generic " curves ).

Suppose X is a normed vector space over R or C. We denote by its continuous dual, i. e. the space of all continuous linear maps from X to the base field.

Suppose that we vary the complex structure of X over a simply connected base.

Suppose that X is a non-singular n-dimensional projective algebraic variety over the field F < sub > q </ sub > with q elements.

Suppose that F is a collection of continuous linear operators from X to Y.

Suppose C is a category, and f: X → Y is a morphism in C. The morphism f is called a constant morphism ( or sometimes left zero morphism ) if for any object W in C and any g, h: W → X, fg

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