Statistics, Probability and Discrete Mathematics — BMATB301 (Modules 1–5): 30 flashcards
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1. What does the method of least squares minimise?
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The sum of squared vertical deviations $\sum (y_i - \hat{y}_i)^2$ between observed and fitted values.
2. How are normal equations obtained in curve fitting?
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By setting the partial derivatives of the error sum $\sum (y_i - \hat{y}_i)^2$ with respect to each unknown coefficient equal to zero.
3. For $y = a_0x^2 + a_1x + a_2$, how many normal equations are needed?
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Three — one for each unknown coefficient $a_0, a_1, a_2$.
4. What is the difference between regression of $y$ on $x$ and regression of $x$ on $y$?
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Regression of $y$ on $x$ predicts $y$ from a known $x$; regression of $x$ on $y$ predicts $x$ from a known $y$. They are different lines.
5. At which point do both regression lines always intersect?
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The point $(\bar{x}, \bar{y})$ — the means of the two variables.
6. State the formula relating the coefficient of correlation to the two regression coefficients.
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$$r = \pm\sqrt{b_{yx} \cdot b_{xy}}$$ with the sign taken from $b_{yx}$.
7. If two regression lines are given as equations, how do you find $\bar{x}$ and $\bar{y}$?
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Solve the two equations simultaneously — their intersection point is $(\bar{x}, \bar{y})$.
8. How do you linearise the power curve $y = ax^b$?
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Take logarithms: $\log y = \log a + b\log x$, which is a straight line in $X = \log x$, $Y = \log y$.
9. State Spearman's rank correlation formula.
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$$\rho = 1 - \frac{6\sum d_i^2}{n(n^2-1)}$$ where $d_i$ is the difference between the two ranks for item $i$.
10. Why is rank correlation robust to outliers?
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Because it uses only the order (ranks) of the data, not the actual magnitudes, so extreme values do not distort it.
11. What condition must a discrete probability distribution satisfy?
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$\sum P(x) = 1$ — the sum of all probabilities must equal 1.
12. When does the Poisson distribution approximate the binomial distribution?
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When $n$ is large and $p$ is small, with $\lambda = np$.
13. What is the density function of the exponential distribution with mean $\theta$?
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$$f(t) = \frac{1}{\theta}e^{-t/\theta}$$
14. What does it mean that the exponential distribution is memoryless?
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Knowing how much time has already elapsed tells you nothing about how much longer you must wait.
15. How do you convert a normal variable $X \sim N(\mu, \sigma^2)$ to standard normal?
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$$z = \frac{X - \mu}{\sigma}$$
16. What is a null hypothesis?
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A statement assumed true until evidence rejects it at a chosen significance level.
17. When is the t-test used instead of the z-test?
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For small samples when the population standard deviation $\sigma$ is unknown.
18. What does the chi-square goodness-of-fit test compare?
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Observed frequencies with expected frequencies under a theoretical distribution or ratio.
19. When is an argument valid?
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When the conclusion must be true whenever all the premises are true.
20. What is a tautology?
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A compound proposition that is true for every possible assignment of truth values to its variables.
21. How does a direct proof of "$n$ even $\Rightarrow$ $n+3$ odd" proceed?
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Write $n = 2k$, then $n+3 = 2k+3 = 2(k+1)+1$, which is odd.
22. What is the strategy of an indirect proof?
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Assume the negation of the conclusion and derive a contradiction.
23. What three properties define a partial order?
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Reflexive, antisymmetric, and transitive.
24. What does a Hasse diagram show?
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Only the covering relations of a poset — no self-loops and no transitive edges.
25. What three properties define an equivalence relation?
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Reflexive, symmetric, and transitive.
26. How does the relation $x - y$ is even partition the integers?
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Into two equivalence classes: even integers and odd integers.
27. State the definition of function composition $(g \circ f)(x)$.
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$(g \circ f)(x) = g(f(x))$ — apply $f$ first, then $g$.
28. How do you verify that two functions $f$ and $g$ are inverses?
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Check that $g(f(x)) = x$ and $f(g(x)) = x$ for all $x$ in the domain.
29. What is the matrix representation $M(R)$ of a relation?
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A matrix with entry 1 when $(a,b) \in R$ and 0 otherwise.
30. How is the digraph of a relation drawn?
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Draw an arrow from $a$ to $b$ whenever $(a,b) \in R$.
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