STAT 400 Chap.5 Continuous Random Variables and the Normal Model
Continuous Random Variables and the Normal Model
Probability becomes area
A continuous random variable is described by a probability density function, a nonnegative curve with total area one. Probabilities are areas under that curve, so any single value has probability zero and strict or non-strict inequalities give the same answer. The density itself is not a probability: it is a rate per unit of x and can exceed one on a narrow interval.
The CDF F(x) is the area to the left of x, its derivative is the density, and P(a ≤ X ≤ b) = F(b) − F(a) turns most questions into a subtraction.
Moments and percentiles by integration
Means and variances replace sums with integrals: μ is the integral of x·f(x), and the variance again uses E[X²] − μ². Calculus II techniques, especially integration by parts for exponential densities, appear here constantly.
The course also asks for percentiles: the 100p-th percentile is the value with area p to its left, found by setting the CDF equal to p and solving.
A median is the 50th percentile.
Three named densities from waiting and spreading
The uniform density spreads probability evenly over an interval, with mean at the midpoint and variance (b − a)²/12. The exponential density models the waiting time to a single event; in the course's parameterisation its mean is θ and its variance θ², and its survival function e^(−x/θ) makes it memoryless, like the geometric distribution.
The gamma density with shape α and scale θ models the wait for the α-th event, with mean αθ and variance αθ². If events follow a Poisson process with rate λ, these waiting times have θ = 1/λ.
The normal model and standardisation
The normal distribution is the most used model in the second half of the course.
It is symmetric about its mean μ with spread set by σ, and any normal variable becomes standard normal after subtracting μ and dividing by σ. That single step means one table of Φ values handles every normal probability. Symmetry gives Φ(−z) = 1 − Φ(z), and percentiles run the standardisation backwards, x = μ + zσ.
The familiar 68, 95 and 99.7 percent figures for one, two and three standard deviations are useful checks on any computed answer.
Looking ahead
The normal model returns as the limiting shape in the central limit theorem and as the reference distribution for z intervals and z tests, so fluency with the table here pays off twice later.
What this chapter covers
- 01
Probability density functions
- 02
CDFs and percentiles
- 03
Continuous expectation and variance
- 04
Uniform and exponential models
- 05
Gamma waiting times
- 06
Normal distribution and z-scores
Worked example · free
Standardise a normal measurement
- 2Standardise the endpoints: z = (35 − 40)/5 = −1.00 and z = (48 − 40)/5 = 1.60.
- 2Use the table: Φ(1.60) − Φ(−1.00) = 0.9452 − 0.1587 = 0.7865.
- 2Top 5%: x = 40 + 1.645(5) ≈ 48.2 hours.
Key terms
- Probability Density Function
- A nonnegative function whose area over an interval gives the probability that a continuous variable falls in that interval.
- Percentile
- The value with a given proportion of the distribution to its left, found by solving F(x) = p.
- Exponential Distribution
- A continuous waiting-time model with density (1/θ)e^(−x/θ) for x > 0, mean θ and the memoryless property.
- Gamma Distribution
- A continuous model for the waiting time to the α-th event of a Poisson process, with mean αθ and variance αθ².
- Standard Normal
- The normal distribution with mean 0 and variance 1, whose CDF Φ is tabulated and used after standardising.
Continuous Random Variables and the Normal Model FAQ
Why is the probability of an exact value zero for a continuous variable?
Probability is area under the density, and a single point has no width, so its area is zero. Only intervals carry positive probability, which is also why including or excluding an endpoint never changes a continuous answer.
How do I find a percentile of a continuous distribution?
Set the CDF equal to the required proportion and solve for x. For a normal variable, look up the matching z value and convert it with x = μ + zσ.
When should I use the exponential rather than the Poisson?
Use the Poisson for the number of events in a fixed interval and the exponential for the waiting time until the next event in the same process.
Exam move
Practise each density with the same three tasks: find a constant, find the CDF, and find a percentile. For the normal model, drill standardisation both forwards and backwards until writing the z-score comes before opening the table.
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