CPA Foundation Level

Quantitative Analysis flashcards

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How are the roots of a x^2 + b x + c = 0 found, and what does the discriminant tell you?

x = [-b + or - sqrt(b^2 - 4ac)] / (2a). The discriminant b^2 - 4ac gives the number of real roots: two if positive, one if zero, none if negative. The turning point (vertex) is at x = -b / 2a: a minimum if a is positive, a maximum if a is negative.

How do you find the determinant and the inverse of a 2 x 2 matrix?

For A = [[a, b], [c, d]]: det A = ad - bc. A^-1 = (1 / det A) x [[d, -b], [-c, a]]: swap the leading-diagonal elements, change the signs of the other two, and divide by the determinant. If det A = 0 the matrix is singular and has no inverse. Check by multiplying back: A x A^-1 = I.

Where is profit maximised, in calculus terms?

Where marginal revenue equals marginal cost (the derivative of profit, MR - MC, is zero) and the second derivative of profit is negative. Example: TR = 100Q - 2Q^2 and TC = 400 + 20Q. MR = 100 - 4Q and MC = 20, so Q = 20; the second derivative of profit, -4, confirms a maximum.

When is the median a better average than the arithmetic mean, and what is the mode?

The mean uses every value but is pulled toward extreme values. The median, the middle value of the ordered data, is unaffected by extremes, so it is the better centre for incomes and property prices. The mode is the value that occurs most often; a distribution may have no mode or more than one.

State the addition and multiplication laws of probability.

Addition: P(A or B) = P(A) + P(B) - P(A and B); for mutually exclusive events, P(A or B) = P(A) + P(B). Multiplication: P(A and B) = P(A) x P(B given A); for independent events, P(A and B) = P(A) x P(B). Complement: P(A prime) = 1 - P(A), the quick route to 'at least one'.

When do you use the binomial distribution and when the Poisson, and what are their means and variances?

Binomial: the number of successes in n independent trials, each with probability p. P(X = r) = nCr x p^r x (1 - p)^(n - r); mean np, variance np(1 - p). Poisson: the number of events in a fixed interval at a constant average rate lambda. P(X = r) = e^(-lambda) x lambda^r / r!; mean and variance both lambda. The Poisson approximates the binomial when n is large and p small, with lambda = np.

What are the steps of every hypothesis test?

1. State H0 (always with an equals sign) and H1 (one-tailed if a direction is asserted, two-tailed if not). 2. Choose the significance level (5% unless told otherwise). 3. Compute the test statistic from the sample. 4. Find the critical value for that statistic, level and tail. 5. Reject H0 if the statistic lies beyond the critical value; otherwise do not reject. 6. State the conclusion in the words of the problem.

What do Pearson's r and Spearman's rank coefficient measure, and when is Spearman preferred?

Pearson's r measures the strength and direction of a linear relationship between two measured variables, from -1 to +1. Spearman's rs = 1 - 6 sum of d^2 / [n (n^2 - 1)] applies the same idea to ranks. Prefer it when the data are rankings, the relationship is monotonic but not linear, or outliers would distort r. Neither shows that one variable causes the other.

What are the four components of a time series?

Trend (T): the long-run, smooth movement of the series. Seasonal variation (S): a pattern that repeats within each year at the same time. Cyclical variation (C): swings above and below the trend over periods longer than a year, linked to the business cycle. Irregular variation (I): the unpredictable residue, which cannot be forecast, only allowed for.

How are EMV and the expected value of perfect information (EVPI) calculated?

EMV of an act = sum over states of P(state) x payoff. Choose the act with the highest EMV. EV with perfect information = sum over states of P(state) x the best payoff in that state. EVPI = EV with perfect information - best EMV, which equals the minimum expected opportunity loss. It is the most worth paying for a perfect forecast.

How is the break-even output of a linear cost and revenue model found?