MATH2089 Chap.9 Random Variables and Distributions
Random Variables and Distributions
Random variables connect uncertain outcomes with numerical models. Discrete variables use probability masses and continuous variables use densities whose interval areas are probabilities. This chapter develops expectation, variance and linear transformations, then compares binomial, Poisson, uniform, exponential and normal distributions through support, parameters and generating mechanisms.
Cumulative distributions and complement rules provide a common route for exact values, ranges, tails and quantiles. A random variable maps outcomes to numbers. Its support lists the values it may take and is the first model check. Discrete support is countable, so probability can be assigned to particular values.
A continuous variable is described by a density; probabilities are areas over intervals and a single point has zero probability even when the density value there is high. Do not read density height as point probability. The cumulative distribution function F(x) equals the probability that X is no greater than x and works for both types.
For a discrete upper tail, strict and non-strict boundaries matter: more than seven differs from at least seven. For a continuous model the endpoint contributes zero probability. Sketch the event and state it symbolically before choosing a mass function, cumulative value or complement. Expectation is the probability-weighted centre of a variable and variance is expected squared distance from that centre.
Adding a constant shifts the mean without changing variance. Multiplying by a constant multiplies the mean by that constant and the variance by its square. Standard deviation then changes by the absolute scale factor. Track units to distinguish variance from standard deviation. Select named distributions from their mechanisms.
A binomial variable counts successes in a fixed number of independent trials with a constant success probability. A Poisson variable counts events over an exposure under an approximately constant rate. A uniform model assigns constant density over an interval. An exponential model describes a memoryless waiting time associated with a constant-rate process.
A normal model is continuous, symmetric and determined by its mean and variance, but a bell-shaped plot alone does not establish the assumptions. Parameter units and exposure must agree. Doubling a Poisson exposure doubles its mean when the rate assumption remains suitable. A normal calculation standardises using the standard deviation, not the variance.
A quantile plot, the generating process and checks for outliers help assess normality. Report the model, parameters, event, numerical probability and contextual interpretation. If an approximation is used, state why it is reasonable and check that its support and boundary conventions match the original variable.
What this chapter covers
- 01
Discrete and continuous variables
- 02
Mass, density and cumulative functions
- 03
Expectation and variance
- 04
Binomial and Poisson models
- 05
Uniform and exponential models
- 06
Normal probabilities and quantiles
Worked example · free
Calculate a binomial probability
- +1Use the exact event X=2.
- +1Count arrangements with the combination 5 choose 2.
- +1Multiply 10 by 0.3 squared and 0.7 cubed.
- +1The result is 0.3087 under the binomial assumptions.
Key terms
- Probability mass
- Probability assigned to a particular value of a discrete random variable.
- Probability density
- A continuous curve whose interval area gives probability.
- Expected value
- A probability weighted average representing the long run centre of a random variable.
Random Variables and Distributions FAQ
Why is a continuous point probability zero?
Probability for a continuous variable is area under a density over an interval. A single point has zero width, even when the density value there is high.
How do binomial and Poisson differ?
A binomial variable counts successes in a fixed number of trials with constant probability. A Poisson variable counts events over an exposure under an approximately constant event rate.
What is the common discrete tail error?
Strict and non-strict integer boundaries differ. For example, fewer than ten means at most nine, so the cumulative function must be evaluated at nine rather than ten.
Exam move
Build a model card for each named distribution with support, parameters, mean, variance and mechanism. Sketch the requested event before using a mass or cumulative function. Practise changing exposure for Poisson models and standardising normal values while keeping variance and standard deviation distinct.
Add one exact-value event and one tail event to every card, noting whether a mass function, cumulative function or complement is required. Use an impossible event to check the support. For binomial questions, state the trials, success definition, constant probability and independence judgement. For Poisson and exponential questions, attach units to rate and exposure.
For normal questions, draw the shaded region before standardising and read a quantile plot beside the mechanism. This routine turns distribution choice into a model decision instead of a formula-recognition exercise.
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