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The first three running averages are 55, 60, and 55. The first average is just the first observation (55).

Conditions: Observations are 55, 65, and 45.; The sample mean is calculated cumulatively.

For any set of distinct positive numbers, the harmonic mean is strictly the smallest, followed by the geometric mean, and the arithmetic mean is the largest. Equality holds only when all input numbers are identical.

Conditions: Inputs are positive real numbers; Comparing H, G, and A means

$S_1$ is the state vector after one iteration of the Markov chain. Its top entry is the probability of being in state A, and its bottom entry is the probability of being in state B after one step.

Conditions: Two-state chain with states A and B.; Entries are probabilities.; In the examples shown, the entries sum to 1.

Total variation in the observed scores is split into two parts: within-group variation, which is caused by differences among individuals inside the same group, and between-group variation, which is associated with differences among the groups themselves.

Conditions: The data are organized into multiple groups.; One is analyzing variation in a measured response across those groups.

To scale an expected value, calculate the expectation for a single unit and then multiply it by the total number of units. This relies on the linearity of expectation, which holds regardless of whether the units are independent, provided the individual expectations are defined.

Conditions: The expectation for a single unit is known or calculable.; The number of units is fixed and finite.

It contradicts minimality because the remainder can be expressed as a linear combination of a and b. If the remainder is positive and smaller than d, it belongs to the set of positive linear combinations, violating the definition of d as the smallest such value.

Conditions: d is defined as the least positive integer linear combination of a and b.; The remainder r is obtained by dividing a or b by d.; r is positive and $r < d$.

For the function $f(x)=\sqrt{x}$, the limit definition of the derivative fails to produce a finite value specifically at $x=0$. While the derivative is $\frac{1}{2\sqrt{x}}$ for all $x>0$, evaluating the limit as $h \to 0^+$ at $x=0$ results in division by zero, indicating an infinite slope (vertical tangent) rather than a defined finite derivative.

Conditions: $f(x)=\sqrt{x}$.; The point of evaluation is $x=0$.; The limit is taken as $h \to 0^+$ (since the domain is $x \ge 0$).

The coefficient $a_j$ is the multiplicity number, which records how many copies of the irreducible representation $\Gamma^j$ occur inside $\Gamma$. If $a_j > 1$, the same irreducible type appears on several different subspaces.

Conditions: The decomposition has been relabeled so that each $j$ corresponds to one inequivalent irreducible representation.

Matrix addition is performed by adding corresponding entries. If two matrices have the same number of rows and columns, their sum is a new matrix where each element $(i,j)$ equals the sum of the elements at position $(i,j)$ in both original matrices.

Conditions: Both matrices must have identical dimensions (same $m \times n$).

The extra assumption required is that the norms of the sequence converge to the norm of the limit, i.e., $\|x_n\| \to \|l\|$. When combined with weak convergence, this condition implies strong convergence.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$ converging weakly to $l$.; $\|x_n\|$ converges to $\|l\|$.

The j-th column of the composed matrix $A \circ B$ is the image under A of the j-th column of B. To find a missing column, identify the corresponding column in B, treat it as an input vector, and apply the transformation A to it.

Conditions: A is a linear transformation represented by a matrix.; Columns are read as vectors in the domain of A.

No, ($A \times B$) ∪ ($C \times D$) generally does not equal (A ∪ C) × (B ∪ D). While the intersection of two Cartesian products is a Cartesian product, their union is typically an irregular shape (like an L-shape) that cannot be expressed as a single Cartesian product of two sets.

Conditions: A, B, C, and D are arbitrary sets.