DDMA Level II
Mathematical Concepts in Data Science flashcards
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Name the five forms of a linear equation and say when each is useful.
General form: Ax + By + C = 0, the most flexible. Slope-intercept form: y = mx + c, for quick graphing. Point-slope form: y - y1 = m(x - x1), when a point and the slope are known. Intercept form: x/a + y/b = 1, where a and b are the x- and y-intercepts. Two-point form: (y - y1)/(y2 - y1) = (x - x1)/(x2 - x1), when two points are known.
How are two vectors added and subtracted, and how is a vector multiplied by a scalar?
Add or subtract corresponding components: (a1, a2) + (b1, b2) = (a1+b1, a2+b2). Scalar multiplication multiplies every component by the same number. With a = (3, 4) and b = (1, 2): a + b = (4, 6), a - b = (2, 2), 2a = (6, 8).
What does the derivative of a function measure?
Its instantaneous rate of change: geometrically, the slope of the tangent line at a point. The derivative as a function gives that rate of change at every point, not just one.
State the power rule and the sum rule of differentiation, and apply them to f(x) = 3x² + 2x.
Power rule: the derivative of xⁿ is n·xⁿ⁻¹. Sum rule: the derivative of a sum is the sum of the derivatives. So f'(x) = 6x + 2, and f'(2) = 6(2) + 2 = 14.
What does the definite integral of f(x) from a to b give, and what is the integral of 2x from 0 to 2?
The exact area under the curve between a and b: the limit of summing thin rectangles as they become infinitesimally narrow. The antiderivative of 2x is x², so the integral is 2² - 0² = 4.
State the addition law of probability and when it applies.
For mutually exclusive events, P(A or B) = P(A) + P(B). Example: on a fair die, P(2 or 5) = 1/6 + 1/6 = 1/3.
State the multiplication law of probability and when it applies.
For independent events, P(A and B) = P(A) × P(B). Example: two fair coins, P(both heads) = 1/2 × 1/2 = 1/4.
What is conditional probability, and what is its formula?
P(A|B) is the probability of event A occurring given that event B has already occurred. P(A|B) = P(A and B) / P(B).
Define the three measures of central tendency.
Arithmetic mean: the sum of all values divided by the number of values. Median: the middle value when the data are arranged in order. Mode: the value that occurs most frequently.
What is the difference between probabilistic and non-probabilistic sampling?
Probabilistic: every unit in the population has a known, non-zero chance of selection, e.g. simple random sampling. Non-probabilistic: selection is not random, e.g. convenience or judgement sampling. Faster, but not statistically representative.
In y = mx + c, what do m and c stand for?
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What is a linear equation in one variable, and how is it solved?
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Solve 2x + y = 10 and x - y = 2 by elimination.
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What are the two methods for solving a system of two linear equations in two unknowns?
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What is a unit vector, and what is the unit vector of a = (3, 4)?
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Which algebraic properties do vector addition and scalar multiplication follow?
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A has rows (2, 1) and (3, 4); B has rows (1, 0) and (2, 5). Find A + B and AB.
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What is the determinant of A, with rows (2, 1) and (3, 4)?
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How is a 2 by 2 matrix inverted, and what is the inverse of A, with rows (2, 1) and (3, 4)?
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Use the inverse of A (rows (0.8, -0.2) and (-0.6, 0.4)) to solve 2x + y = 5 and 3x + 4y = 10.
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What is calculus, and what are its two branches?
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What is a limit, and when is a function continuous?
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What are the three ways of evaluating a limit?
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What is a function, and what are its domain and range?
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State the product rule of differentiation.
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State the chain rule of differentiation.
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Find the minimum of f(x) = x² - 4x + 7 and show that it is a minimum.
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Total cost in thousands of shillings is q³ - 6q² + 15q + 20 for q units. What is the marginal cost at q = 2?
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How is the area under a curve approximated before the definite integral is used?
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What does the fundamental theorem of calculus state?
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How does the power rule work when differentiating, and when integrating?
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Evaluate the integral of 2x + 3 from 0 to 2, and say what an integral means.
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Define experiment, outcome, sample space and event.
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What is the difference between mutually exclusive and mutually inclusive events?
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What is the difference between dependent and independent events?
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How does drawing with or without replacement change the events?
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A bag holds 5 red and 3 blue balls. Two are drawn without replacement. What are P(first red) and P(second red | first red)?
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What does a probability tree diagram show, and how is a combined outcome's probability found?
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What is a finite probability space?
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5% of items from a line are defective, independently. What is the probability that both of two items are defective, and that at least one is?
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A test detects a condition 90% of the time, gives a false positive 5% of the time, and 2% of people have the condition. What is the probability that a person who tests positive has it?
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What is the difference between primary and secondary data, and how is primary data collected?
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What do a frequency table and a histogram show?
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Define the range, the mean deviation and the standard deviation.
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Why does the standard deviation weigh large deviations more heavily than the mean deviation?
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For the data 2, 4, 4, 6, 9, find the mean, median, mode, range, mean deviation and standard deviation.
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Five daily downtimes are 4, 8, 6, 5 and 7 hours. Find the sample variance and sample standard deviation.
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