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The difference is that the first average regards each test as having equal weight, while the second average regards each question as having equal weight. If it really doesn't matter how many questions are on a test, then the first average is better. If it really doesn't matter how the question pool is divided up into tests, then the second average is ...

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First, random and stochastic are synonyms. A stochastic process is a collection of random variables $X=(X_i)_{i\in I}$. You have two important cases: the discrete case: $I=\mathbb{N}$ or $\mathbb{Z}$ (or a subset of those). Then the process is a sequence of random variables. the continuous case : $I=\mathbb{R}$ or $[0,+\infty )$ for example. Then the ...

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Each $\mathbf q_i$ is an attitude estimate ("in quaternion form," so I'd say it is a quaternion). $A(\mathbf{q})$ is a matrix. More specifically, the paper says on page 3, it is the attitude matrix of a quaternion $\mathbf{q}$. The fact that the equation doesn't have a summation and the use of an arbitrary index $i$ also tell you that the expression is ...

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You can: Keep track of the total number of values you have seen so far, call it $n$, and the average of the numbers so far, $A$. Then when you see a new number $x$, map \begin{align*} n &\mapsto n+1 \\ A &\mapsto \frac{nA + x}{n+1} \end{align*} Or: Keep track of the total number of values you have seen so far, call it $n$, and the sum of the ...

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$r_X[k]$ is auto-correlation of WSS random process $X[k]$ given by $$r_X[k]=\frac{E[(X[t]-\mu)(X[t+k]-\mu)]}{\sigma^2}$$ where $\mu=E[X[t]]$ and $\sigma^2=D(X[t])$ are expectation and variance of random process $X$. In this case, sample mean $\hat{\mu}_N=A$ for all $N$ and the sequence $A,A,\ldots$ almost surely does not converge to $\mu$ (it converges to ...

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