Distributions & Random Variables | IB Math AAHL | Perplex

Distributions & Random Variables

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Watch comprehensive video reviews for Distributions & Random Variables, designed for final exam preparation. Each video includes integrated problems you can solve alongside detailed solutions.

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Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.

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Not your average video:

Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.

Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.

Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.

HL

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Normalization

AHL 4.14

By the law of total probability

P(−∞≤X≤∞)=∫−∞∞f(x)dx=1🚫

We say the that probability density function is normalized.

If a probability density function is only defined for an interval a≤X≤b, then it is zero everywhere else and the integral

∫−∞∞f(x)dx=∫abf(x)dx=1

Example

The continuous random variable X has the probability density function f(x)=k(4−x2), for −2≤x≤2. Find the value of k.

Integrating over all possible values

P(−2≤X≤2) 1 1 1=k∫−224−x2dx=1 =k[4x−3x3]−22 =k[(8−38)−(8+38)] =k⋅332⇒k=323

Normalization

AHL 4.14

By the law of total probability

P(−∞≤X≤∞)=∫−∞∞f(x)dx=1🚫

We say the that probability density function is normalized.

If a probability density function is only defined for an interval a≤X≤b, then it is zero everywhere else and the integral

∫−∞∞f(x)dx=∫abf(x)dx=1

Example

The continuous random variable X has the probability density function f(x)=k(4−x2), for −2≤x≤2. Find the value of k.

Integrating over all possible values

P(−2≤X≤2) 1 1 1=k∫−224−x2dx=1 =k[4x−3x3]−22 =k[(8−38)−(8+38)] =k⋅332⇒k=323