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MH HSC 12 Maths

Probability Distributions

Each concept is a short lesson, then questions until you get three right.

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7 concepts, a short lesson each

  • Discrete random variables0/4
  • Continuous random variables0/3

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Random variables

A random variable gives a number to each outcome of an experiment; it is discrete when its values can be listed, continuous when they fill an interval

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Discrete random variables0 of 4

  1. Random variablesA random variable gives a number to each outcome of an experiment; it is discrete when its values can be listed, continuous when they fill an intervalStart
  2. Probability mass functionA p.m.f. lists P(X=x)P(X = x) for each value: every probability is between 00 and 11 and they add to 11; use the total to find an unknown kkStart
  3. Writing a probability distributionList every value the random variable can take, find the probability of each from the experiment, and check that they add to 1Start
  4. Expected value and varianceE(X)=∑x P(x)E(X) = \sum x\,P(x); Var(X)=∑x2P(x)−[E(X)]2\text{Var}(X) = \sum x^2P(x) - [E(X)]^2; the standard deviation is Var(X)\sqrt{\text{Var}(X)}Start

Continuous random variables0 of 3

  1. Probability density functionA p.d.f. has f(x)≥0f(x) \ge 0 and ∫f(x) dx=1\int f(x)\,dx = 1 over its range; P(a<X<b)=∫abf(x) dxP(a < X < b) = \int_a^b f(x)\,dx, the area under the curveStart
  2. Cumulative distribution functionF(x)=P(X≤x)F(x) = P(X \le x): a running total of the p.m.f. for a discrete variable, ∫−∞xf(t) dt\int_{-\infty}^{x} f(t)\,dt for a continuous oneStart
  3. Expected value of a continuous random variableE(X)=∫x f(x) dxE(X) = \int x\,f(x)\,dx and Var(X)=∫x2f(x) dx−[E(X)]2\text{Var}(X) = \int x^2 f(x)\,dx - [E(X)]^2, with integrals in place of the sumsStart

Practise Probability Distributions

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions