IB Maths AI 4.10 Definitions

This page contains our IB Maths AI definitions for 4.10. By learning each one of these definitions, you will fully cover the content for IB Maths AI 'Linear transformations & combinations'.

All Topic 4 Sub-topics 4.1: Fundamentals Of Statistics 4.2: Visualising Statistics 4.3: Distribution Properties 4.4: Introduction To Linear Statistics 4.5: Probability 4.6: Probability Distributions 4.7: Hypotheses & Tests (HL)4.8: Data collection & categorisation (HL)4.9: Advanced linear statistics (HL)4.10: Linear transformations & combinations (HL)4.11: Confidence intervals (HL)4.12: Poisson distribution (HL)4.13: Distribution tests (HL)4.14: Transition matrices

central limit theorem

Stating that for large '

n

', the sample mean '

\bar{X}

' is approximately normally distributed with '

\bar{X}\approx N\left(\mu,\frac{\sigma^2}{n}\right)

' even when the population distribution is not normal.

estimate

A sample-based value used to approximate an unknown population parameter such as '

\mu

' or '

\sigma^2

independent probability

Describes events or random variables where knowing one outcome gives no information about the other; for Poisson variables this condition allows their totals to remain Poisson.

linear combinations

A weighted sum of random variables such as '

a_1X_1+a_2X_2+\dots+a_nX_n

', whose expected value is the corresponding weighted sum of expected values.

linear transformation

Changing a random variable '

X

' to '

aX+b

', which shifts the mean by '

b

' and scales it by '

a

', while scaling the variance by '

a^2

population

The full set of all possible values of interest, described by parameters such as the mean '

\mu

' and variance '

\sigma^2

sample

A set of '

n

' observed values taken from a population and used to compute statistics such as '

\bar{x}

' and '

s_{n-1}^2

sample mean

The arithmetic average of the sample values, given by '

\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i

', and used as an unbiased estimator of '

\mu

sample variance

A statistic measuring spread based on squared deviations from '

\bar{x}

', commonly computed as '

s_{n-1}^2=\frac{1}{n-1}\sum_{i=1}^{n}\left(x_i-\bar{x}\right)^2

' to estimate '

\sigma^2

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