Any other operation on untyped constants results in an untyped constant of the same kind; that is, a boolean, integer, floating-point, complex, or string constant. If the untyped operands of a binary operation (other than a shift) are of different kinds, the result is of the operand's kind that appears later in this list: integer, rune, floating-point, complex. For example, an untyped integer constant divided by an untyped complex constant yields an untyped complex constant.
Applying the built-in function
complex
to untyped integer, rune, or floating-point constants yields an untyped complex constant.
Constant expressions are always evaluated exactly; intermediate values and the constants themselves may require precision significantly larger than supported by any predeclared type in the language. The following are legal declarations:
The divisor of a constant division or remainder operation must not be zero:
The values of typed constants must always be accurately representable by values of the constant type. The following constant expressions are illegal:
The mask used by the unary bitwise complement operator
^
matches the rule for non-constants: the mask is all 1s for unsigned constants and -1 for signed and untyped constants.
Implementation restriction: A compiler may use rounding while computing untyped floating-point or complex constant expressions; see the implementation restriction in the section on
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. This rounding may cause a floating-point constant expression to be invalid in an integer context, even if it would be integral when calculated using infinite precision, and vice versa.
At package level,
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determine the evaluation order of individual initialization expressions in
variable declarations
. Otherwise, when evaluating the
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of an expression, assignment, or
return statement
, all function calls, method calls, and communication operations are evaluated in lexical left-to-right order.
For example, in the (function-local) assignment
the function calls and communication happen in the order
f()
,
h()
,
i()
,
j()
,
<-c
,
g()
, and
k()
. However, the order of those events compared to the evaluation and indexing of
x
and the evaluation of
y
is not specified.
At package level, initialization dependencies override the left-to-right rule for individual initialization expressions, but not for operands within each expression:
The function calls happen in the order
u()
,
sqr()
,
v()
,
f()
,
v()
, and
g()
.
Floating-point operations within a single expression are evaluated according to the associativity of the operators. Explicit parentheses affect the evaluation by overriding the default associativity. In the expression
x + (y + z)
the addition
y + z
is performed before adding
x
.
has a t-distribution with (n-1) d.f. For large sample size (say, n over 30), the new
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has an expected value equal to zero, and its variance is (n-1)/(n-3) which is close to one.
Notice that the t- statistic is related to F-statistic as follow: F = t, where F has (d.f. = 1, and d.f. = d.f. of the t-table)
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to obtain its P-values.
The parameters for the triangular distribution are Minimum (a), Maximum (b), and Likeliest (c). There are three conditions underlying triangular distribution:
These three parameters forming a triangular shaped distribution, which shows that values near the minimum and maximum are less apt to occur than those near the most likely value.
The following are the general Triangular density function, together with the expected value and the variance for a Triangular random variable X (a, c, b):
f(x) = 2(x-a) / [(b-a)(c-a)], for a x c f(x) = 2(b-x) / [(b-a)(b-a)], for c x b E(X) = (a + b + c) / 3 Var(X) = (a + b + c - ab - ac - bc) / 18
Given X is distributed as above, compute the tails probability P (X 0.1 OR X 0.9).
Used to generate random numbers in sampling and Monte Carlo simulation.
Comments: Special case of beta distribution.
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and performing some numerical experimentation for a deeper understanding of the concepts.
Notice that any Uniform distribution has uncountable number of modes having equal density value; therefore it is considered as a homogeneous population.
The discrete uniform distribution describes the distribution of n equally likely events (labeled with the integers from 1 to n), each with probability 1/n.
If X is a discrete uniform random variable with parameter n, then the mean, and variance are as follows:
E(X) = (n+1)/2, Var(X) = (n -1) /12
For two populations use the F-test. For 3 or more populations, there is a practical rule known as the"Rule of 2". In this rule, one divides the highest variance of a sample by the lowest variance of the other sample. Given that the sample sizes are almost the same, and the value of this division is less than 2, then the variations of the populations are almost the same.
Notice: This important condition in analysis of variance (ANOVA and the t-test for mean differences) is commonly tested by the Levene test or its modified test known as the Brown-Forsythe test. Interestingly, both tests rely on the homogeneity of variances condition!
Further Readings: Good Ph., and J. Hardin, Common Errors in Statistics , Wiley, 2003. Wang H., Improved confidence estimators for the usual one-sided confidence intervals for the ratio of two normal variances, Statistics Probability Letters , Vol. 59, No.3, 307-315, 2002.