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Machine learning Interview > Probability > Flashcards

Probability Flashcards

(84 cards)

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1
Q

What is a Random Variable?

A

A variable that can take on different values determined by the outcomes of a random phenomenon.

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2
Q

What are the two types of Random Variables?

A

Discrete and Continuous.

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3
Q

What is a Discrete Probability Distribution?

A

A distribution that shows the probabilities of outcomes of a discrete random variable.

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4
Q

Give an example of a Discrete Probability Distribution.

A

The probability of getting heads or tails when flipping a coin.

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5
Q

What is a Binomial Distribution?

A

A distribution representing the number of successes in a fixed number of independent Bernoulli trials.

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6
Q

What are the key parameters of a Binomial Distribution?

A

Number of trials (n) and probability of success (p).

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7
Q

What is the Binomial Coefficient used for?

A

To calculate the number of possible combinations in a binomial experiment.

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8
Q

What is the formula for the Binomial Coefficient?

A

C(n, k) = n! / (k!(n-k)!)

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9
Q

What is a Bernoulli Distribution?

A

A discrete probability distribution for a random experiment with only two outcomes: success (1) and failure (0).

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10
Q

What is the expected value of a Bernoulli Distribution?

A

The probability of success (p).

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11
Q

What is a Continuous Probability Distribution?

A

A distribution that shows the probabilities of outcomes for a continuous random variable.

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12
Q

What is a Probability Density Function (PDF)?

A

A function that describes the likelihood of a continuous random variable taking a particular value.

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13
Q

What is the property of a PDF?

A

The area under the curve equals 1.

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14
Q

What is a Cumulative Distribution Function (CDF)?

A

A function that gives the probability that a random variable is less than or equal to a certain value.

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15
Q

What is the relationship between PDF and CDF?

A

The CDF is the integral of the PDF.

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16
Q

What is a Uniform Distribution?

A

A distribution where all outcomes are equally likely within a given interval.

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17
Q

What is the PDF formula for a Uniform Distribution?

A

1 / (b - a) for values between a and b.

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18
Q

What is a Normal Distribution?

A

A symmetric, bell-shaped distribution defined by its mean (μ) and standard deviation (σ).

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19
Q

What rule is often associated with Normal Distributions?

A

The Empirical Rule (68-95-99.7 rule).

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20
Q

What is a Chi-Squared Distribution?

A

A distribution of a sum of squared standard normal variables, often used in hypothesis testing.

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21
Q

What is Sampling from a Distribution?

A

Selecting a subset of data points from a probability distribution to estimate population properties.

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22
Q

What are the two common methods of Sampling?

A

Random Sampling and Stratified Sampling.

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23
Q

What is Probability?

A

A measure of the likelihood that a particular event will occur, ranging from 0 to 1.

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24
Q

What is the formula for Probability?

A

Probability = Number of favorable outcomes / Total number of possible outcomes.

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25
What is the probability of rolling a 4 on a fair six-sided die?
36897
26
What is the Complement of a Probability?
The probability that an event does not occur. P(Not A) = 1 - P(A)
27
What is the Sum Rule for Disjoint (Mutually Exclusive) Events?
The probability of one or another occurring is the sum of their individual probabilities.
28
What is the Sum Rule for Joint Events?
The probability of either event A or B occurring is P(A) + P(B) - P(A and B).
29
What does it mean if two events are Independent?
The occurrence of one event does not affect the probability of the other.
30
What is the Birthday Problem?
A probability puzzle that finds the chance that, in a group of people, at least two will share the same birthday.
31
What is Conditional Probability?
The probability of one event occurring given that another has already occurred.
32
What is the formula for Conditional Probability?
P(A|B) = P(A and B) / P(B)
33
What is Bayes' Theorem?
A formula that describes how to update the probabilities of hypotheses when given evidence.
34
What is the formula for Bayes' Theorem?
P(A|B) = [P(B|A) * P(A)] / P(B)
35
What is a Prior Probability in Bayes' Theorem?
The initial probability of an event before new evidence is considered.
36
What is a Posterior Probability in Bayes' Theorem?
The updated probability of an event after taking new evidence into account.
37
What is the Naive Bayes Model?
A probabilistic classifier based on Bayes' Theorem assuming independence between features.
38
How is Bayes' Theorem applied in spam filtering?
By computing the probability that an email is spam given the presence of certain words or phrases.
39
What is the Monty Hall Problem?
A probability puzzle involving choosing one of three doors, with a twist that affects your winning chances if you switch your choice.
40
How is Probability applied in Machine Learning?
In modeling uncertainty, predictions, classifiers like Naive Bayes, and in algorithms involving randomness.
41
What is Expected Value?
The long-run average value of repetitions of a random experiment. For a discrete random variable, it is the weighted average of all possible outcomes.
42
How is the median defined?
The middle value in a sorted list of numbers, separating the higher half from the lower half.
43
What is the mode?
The value that appears most frequently in a dataset.
44
What is the Expected Value of a Function?
For a random variable X and function g, E[g(X)] = ∑ g(x) * P(X=x) (discrete) or ∫ g(x) * f(x) dx (continuous).
45
What is the Sum of Expectations rule?
For any random variables X and Y, E[X + Y] = E[X] + E[Y], regardless of independence.
46
How is Variance defined?
A measure of how far a set of numbers are spread out from their mean: Var(X) = E[(X - μ)²].
47
What is Standard Deviation?
The square root of the variance, providing a measure of dispersion in the same units as the data.
48
What is the Sum of Gaussians property?
The sum of independent Gaussian random variables is also Gaussian.
49
What does Standardizing a Distribution mean?
Transforming data to have a mean of 0 and a standard deviation of 1: Z = (X - μ)/σ.
50
What are Skewness and Kurtosis?
Skewness measures asymmetry; Kurtosis measures tail heaviness compared to a normal distribution.
51
What does Skewness indicate?
Positive skew = right-tailed; Negative skew = left-tailed. Measures lack of symmetry.
52
What does Kurtosis indicate?
High kurtosis = heavy tails/sharp peak; Low kurtosis = light tails/flat peak.
53
What are Quantiles?
Values dividing the data into equal-sized intervals (e.g., quartiles divide into quarters).
54
What is a Box-Plot?
A graphical display showing data distribution via quartiles, median, and outliers.
55
What is Kernel Density Estimation?
A non-parametric way to estimate the probability density function of a random variable.
56
What is a Violin Plot?
A combination of a box plot and kernel density plot, showing distribution shape.
57
What is a QQ Plot?
Quantile-Quantile plot, comparing two distributions by plotting their quantiles against each other.
58
What is a Joint Distribution (Discrete)?
The probability distribution of two or more discrete random variables.
59
What is a Joint Distribution (Continuous)?
The probability distribution of two or more continuous random variables.
60
What are Marginal and Conditional Distributions?
Marginal: distribution of a subset ignoring others. Conditional: distribution given another variable's value.
61
What is Covariance in a Dataset?
A measure of how much two random variables change together: Cov(X,Y) = E[(X-μₓ)(Y-μᵧ)].
62
What is the Covariance of a Probability Distribution?
The expected value of the product of deviations from the mean for two random variables.
63
What is a Covariance Matrix?
A matrix where each element (i,j) is the covariance between the i-th and j-th random variables.
64
What is the Correlation Coefficient?
A standardized measure of linear dependence: ρ = Cov(X,Y)/(σₓσᵧ), ranging from -1 to 1.
65
What is a Confidence Interval (CI)?
A range of values, derived from sample data, that is likely to contain the true population parameter with a specified confidence level (e.g., 95%).
66
How do you interpret a 95% CI?
'We are 95% confident that the true parameter lies within this interval.' (Note: The parameter is fixed; the interval is random.)
67
What factors affect the width of a CI?
1. Sample size (↑n → ↓width), 2. Confidence level (↑confidence → ↑width), 3. Variability (↑σ → ↑width).
68
What is the Margin of Error (MoE)?
The radius of the CI: MoE = Critical Value × Standard Error. For 95% CI (Z=1.96): MoE = 1.96 × (σ/√n).
69
How do you calculate a CI for a mean (σ known)?
CI = X̄ ± Z*(σ/√n), where Z is the critical value (e.g., 1.96 for 95% CI).
70
How do you calculate a CI for a mean (σ unknown)?
Use the t-distribution: CI = X̄ ± t*(s/√n), where t depends on degrees of freedom (n-1).
71
What is the difference between confidence and probability?
Confidence refers to long-run frequency (e.g., 95% of CIs will contain the true parameter). Probability is about a single event.
72
How do you calculate a CI for a proportion?
CI = p̂ ± Z*√(p̂(1-p̂)/n), where p̂ is the sample proportion.
73
What sample size is needed for a desired MoE?
n = (Z² × σ²) / MoE² (for means) or n = (Z² × p(1-p)) / MoE² (for proportions).
74
What are Type I and Type II errors?
Type I (False Positive): Rejecting a true H₀. Type II (False Negative): Failing to reject a false H₀.
75
What is the p-value?
The probability of observing data as extreme as the sample, assuming H₀ is true. Small p-value → reject H₀.
76
What are critical values?
Thresholds in a test statistic's distribution that define rejection regions (e.g., Z=1.96 for α=0.05 two-tailed).
77
What is the power of a test?
Probability of correctly rejecting H₀ (1 - Type II error). Increases with effect size, sample size, and α.
78
What is a t-test?
A test comparing means using the t-distribution (used when σ is unknown or n < 30). Types: One-sample, two-sample, paired.
79
When do you use a two-sample t-test?
To compare means of two independent groups (e.g., treatment vs. control). Assumes equal variances (or use Welch’s test).
80
What is a paired t-test?
Compares means of the same group under two conditions (e.g., before/after). Uses differences between pairs.
81
What is A/B testing?
A statistical method to compare two versions (A/B) to determine which performs better (e.g., webpage conversions).
82
How do you test proportions?
Use a z-test: z = (p̂₁ - p̂₂) / √(p̂(1-p̂)(1/n₁ + 1/n₂)), where p̂ is the pooled proportion.
83
What is the t-distribution?
A symmetric, bell-shaped distribution with heavier tails than Normal. Approaches Normal as n → ∞.
84
What are right-tailed, left-tailed, and two-tailed tests?
Right-tailed: H₁ > H₀. Left-tailed: H₁ < H₀. Two-tailed: H₁ ≠ H₀. Determines rejection region direction.

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