STAT1008 Chap.9 Normal and Uniform Distributions
Normal and Uniform Distributions
Normal and Uniform Distributions moves from tables to densities, where probability is area and any single value has probability zero. It standardises normal variables to Z, solves forward problems from a value to a probability and inverse problems from a percentage to a cut-off, and checks normality with a histogram and a normal probability plot.
The uniform model is handled with rectangles, and the normal approximation to the binomial uses a continuity correction. Assessment questions in this area combine a sketch, a standardisation and a table look-up, often in both directions within one problem. Later parts may chain distributions, using a normal or uniform probability as the success rate of a binomial count that is then approximated by a normal curve.
What this chapter covers
- 01
Density curves and probability as area
- 02
Properties of the normal distribution
- 03
Standardising to Z and the cumulative table
- 04
Inverse normal problems
- 05
Assessing normality
- 06
The uniform distribution
- 07
Normal approximation to the binomial
Worked example · free
Probability for a uniform delivery window
- 1X ~ Unif(10, 40), so the density height is 1/30.
- 1P(15 < X < 25) = (25 − 15)/30 = 0.3333.
- 1E(X) = (10 + 40)/2 = 25 minutes.
- 1SD(X) = (40 − 10)/√12 = 8.66 minutes.
Key terms
- Probability density function
- A probability density function is a curve whose area over an interval gives the probability of that interval.
- Standard normal distribution
- The standard normal distribution is the normal distribution with mean 0 and standard deviation 1.
- Continuity correction
- A continuity correction moves each boundary by one half when a normal curve approximates a discrete count.
- Uniform distribution
- A uniform distribution assigns equal probability to every interval of the same length within its range.
- Z-value
- A z-value is a point on the standard normal scale, found by subtracting the mean and dividing by the standard deviation.
- Inverse normal problem
- An inverse normal problem starts from a probability and finds the value of the variable that cuts it off.
Normal and Uniform Distributions FAQ
Why is the probability of an exact value zero for a continuous variable?
Probability for a continuous variable is area under the density curve, and a single point has no width, so its area is zero. That is also why P(X less than a) and P(X at most a) are equal for a continuous variable.
How do I find a cut-off such as the top 10 per cent?
Convert the wording to a lower-tail area, here 0.90, find the z-value with that cumulative probability, then use x equals the mean plus z times the standard deviation. Excel’s NORM.INV does the same in one step.
When can a binomial be approximated by a normal curve?
The course’s condition is that np and n(1 minus p) both exceed 5. Use a normal curve with mean np and standard deviation the square root of np(1 minus p), and apply the continuity correction when the exam asks for it.
How do I read the cumulative normal table for an upper tail?
The table gives the probability below a z-value. For the probability above it, subtract the table value from 1. By symmetry, the area above a positive z equals the area below the matching negative z, which saves a look-up.
How do I check whether data are approximately normal?
Start with a histogram and look for a roughly symmetric bell shape. Then draw a normal probability plot of the ordered data against standard normal quantiles; points lying close to a straight line support a normal model.
Exam move
Sketch a normal curve and shade the region before every calculation, then standardise and read the table. Alternate forward and inverse questions so you never confuse the direction, and add one problem with two unknowns each session. Learn the uniform mean and variance formulas side by side with their normal counterparts, and practise one chained question that moves from a normal or uniform probability to a binomial count.
As a final check, explain aloud why the probability of an exact value is zero; if the explanation is clear, the area idea is secure. Keep a two-column page of forward and inverse problems with the same numbers, so you see how each direction rearranges the same standardising equation.
Finish each session with one normal approximation to a binomial, writing the np and n(1 minus p) check, the approximating distribution and the corrected boundary explicitly.
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