Continuous random variables and the normal distribution
Continuous random variables and the normal distribution in IB Analysis and Approaches HL: density, areas, inverse normal and z-scores.
When a variable can take any value in an interval (a height, a time, a weight), it is continuous and you no longer assign a probability to each value, but to intervals: the probability of an interval is the area under a curve, the probability density function. In this topic you learn to work with any density function (at HL, using integrals) and with the most important one, the normal distribution: its bell curve, calculating probabilities with the graphic display calculator, the inverse normal (which value leaves a given probability) and standardisation to find the mean or the standard deviation. In the exam you almost always have to set up the model, draw the area and interpret the result in context.
Recommended order
Work through the sections in order: continuous random variables and density functions, the normal distribution, normal calculations and the inverse normal and standardisation and finding parameters. At the end you have the IB exam questions grouped by type.
Discrete random variables (topic 15), definite integrals and standard deviation (topics 11 and 12).
Sections in this topic
Continuous random variable and probability density function
What a continuous random variable is, how a probability is found as an area under the density function, and how to calculate mean, variance, median and mode.
3 worked examples Section 2The normal distribution
What the normal distribution N(μ, σ²) is, its bell curve, what μ and σ do, and how to use the 68-95-99.7 rule and symmetry. IB Analysis and Approaches HL.
3 worked examples Section 3Normal calculations and the inverse normal
Normal probabilities on a GDC, finding the value for a given probability (inverse normal) and solving context problems. IB Analysis and Approaches HL.
3 worked examples Section 4Standardisation and finding μ and σ
What the standardised variable Z is, how to compare positions with z-values and how to find the mean and standard deviation of a normal from probabilities.
3 worked examples End of topicIB exam questions: topic 16
IB exam questions on continuous random variables and the normal distribution: density, probabilities, inverse normal, with solutions.
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