International Baccalaureate Mathematics

Syllabus Content

Variance of a discrete random variable.
Continuous random variables and their probability density functions.
Mode and median of continuous random variables.
Mean, variance and standard deviation of both discrete and continuous random variables.
The effect of linear transformations of X.

Here are some exam-style questions on this statement:

"A continuous random variable $X$ has probability density function $f$ defined by" ... more

"The continuous random vanable $X$ has probability density function:" ... more

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Probability Probability is a measure of the weight of evidence, and is arrived at through reasoning and inference. In simple terms it is a measure or estimation of likelihood of the occurrence of an event. The word probability comes from the Latin word probabilitas which is a measure of the authority of a witness in a legal case. Some of the earlier mathematical studies of probability were motivated by the desire to be more profitable when gambling. Today however the practical uses of probability theory go far beyond gambling and are used in many aspects of modern life. We believe that even adults can, in many cases, have a poor intuition regarding the effects of probability. These activities are designed to help pupils calculate but also get a 'feel' for the principles of probability.

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Official Guidance, clarification and syllabus links:

Link to: discrete random variables (SL 4.7)

$0 \le f(x) \le 1,\int_{-\infty}^{\infty} f(x) dx=1$ including piecewise functions.

For a continuous random variable, a value at which the probability density function has a maximum value is called a mode and for the median:

$ \int_{-\infty}^{m} f(x) dx= \frac{1}{2} $

Use of the notation $E(X), E(X^2),Var(X), \text{ where } Var(X)=E(X^2)-[E(X)]^2 $ and related formulae.

Use of $E(X) $ for "fair" games.

$ E(aX+b)=aE(X)+b, Var(aX+b)=a^2Var(X) $

Formula Booklet: Formulae

The variance of a discrete random variable measures the spread or dispersion of the probability distribution. Mathematically, if $ X $ is a discrete random variable with mean $ \mu $ and probability mass function $ p(x) $, the variance $ \sigma^2 $ is given by: $$ \sigma^2 = \sum (x - \mu)^2 \cdot p(x) $$

For continuous random variables, we use probability density functions (pdfs) instead of probability mass functions. The pdf, denoted by $ f(x) $, gives the likelihood of the variable taking on a particular value. The total area under the curve of a pdf is always 1. The mode of a continuous random variable is the value of $ x $ for which the pdf has its maximum value. The median is the value $ m $ such that the area under the curve to the left of $ m $ is 0.5.

The mean (or expected value) of both discrete and continuous random variables is a measure of the central tendency. For a discrete random variable $ X $ with probability mass function $ p(x) $, the mean $ \mu $ is: $$ \mu = \sum x \cdot p(x) $$ For a continuous random variable with pdf $ f(x) $, the mean is: $$ \mu = \int_{-\infty}^{\infty} x \cdot f(x) \, dx $$ The variance and standard deviation for continuous random variables can be found using similar formulas, but with integration instead of summation.

When we apply a linear transformation of the form $ Y = aX + b $ to a random variable $ X $, the mean and variance change in predictable ways. The new mean $ \mu_Y $ and variance $ \sigma_Y^2 $ are given by: $$ \mu_Y = a\mu_X + b \\ \sigma_Y^2 = a^2\sigma_X^2 $$ The standard deviation $ \sigma_Y $ of $ Y $ is $ |a| $ times the standard deviation $ \sigma_X $ of $ X $.

Not random, not variable - from Twitter

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International Baccalaureate Mathematics

Statistics and Probability

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