DPST1012 Chap.12 Probability Distributions and Statistics
Probability Distributions and Statistics
Distributions of random variables
Topic 9 in Week 11 describes random variables by their distributions, and Topic 10 in Week 12 adds statistics before revision. For a discrete variable, the syllabus asks for the distribution as a function, a table and a graph, and for its expected value, variance, standard deviation, median and mode.
The binomial distribution has its own mean np, variance np(1 − p) and a skewness that depends on whether p is below, at or above one half.
Continuous variables and the Normal
A continuous variable is described by a probability density function, and probabilities are areas under it, so a single value has probability zero.
The Normal distribution is the bell-shaped, symmetric case, and every Normal question reduces to the standard Normal through the z-score, the number of standard deviations a value lies from the mean.
A normal distribution table is provided in the final exam, so reading it with symmetry and subtraction is the core skill, together with working backwards from a probability to a value.
Correlation, regression and testing
The last topic introduces three ideas.
Pearson's correlation coefficient measures the strength and direction of a linear relationship on a scale from −1 to 1. A linear regression line models real data and gives predictions, which are trustworthy inside the range of the data and doubtful outside it. Hypothesis testing asks whether data are surprising under a null hypothesis and reaches a conclusion stated in the context of the question.
What this chapter covers
- 01
Discrete distributions as table, function and graph
- 02
Expected value, variance and standard deviation
- 03
Median and mode of a distribution
- 04
The binomial distribution and skewness
- 05
Continuous distributions and the PDF
- 06
The Normal distribution and z-scores
- 07
Correlation and linear regression
- 08
Introduction to hypothesis testing
Worked example · free
Normal probabilities in both directions
- 1z = (47 − 40)/5 = 1.4.
- 2From the table, P(Z < 1.4) = 0.9192, so P(lifetime < 47) ≈ 0.919.
- 1The top 5% begins where P(Z < z) = 0.95, at z ≈ 1.645.
- 2x = μ + zσ = 40 + 1.645 × 5 ≈ 48.2 hours.
Key terms
- Expected Value
- The long-run average of a random variable, the sum of each value times its probability.
- Standard Deviation
- The square root of the variance, measuring the typical distance of values from the mean.
- Normal Distribution
- The continuous, symmetric, bell-shaped distribution described by its mean and standard deviation.
- Z-Score
- The number of standard deviations a value lies above or below the mean, found as (x − μ)/σ.
- Correlation Coefficient
- Pearson's r, a number from −1 to 1 measuring the strength and direction of a linear relationship.
- Null Hypothesis
- The default claim tested in a hypothesis test, kept unless the data are sufficiently surprising.
Probability Distributions and Statistics FAQ
Why is P(X = a) zero for a continuous variable?
Probability for a continuous variable is area under the density curve, and the area above a single point has no width. So only intervals carry probability, and P(X < a) equals P(X ≤ a).
How do I use a normal table for a negative z-score?
Use symmetry. The table gives P(Z < z) for positive z, and P(Z < −z) equals 1 − P(Z < z). For an interval, subtract the two cumulative values at its ends.
Does a strong correlation prove that one variable causes the other?
No. A strong correlation shows that the two variables move together in a linear way, but a third factor or coincidence can produce that pattern. Causation needs a designed experiment or other evidence.
What should the conclusion of a hypothesis test look like?
A sentence in context that says whether there is evidence against the null hypothesis at the stated significance level, such as whether the mean appears to have changed, rather than a bare reject or accept.
What makes a table a valid probability distribution?
Every probability must lie between 0 and 1, and together they must add to exactly 1. If one entry is unknown, that total gives the equation to find it, and a negative result means the table cannot be a distribution.
Exam move
Build one discrete distribution table a week and compute its mean and variance by hand, then answer four Normal questions with a printed table, two forwards and two backwards. Revise this chapter last, closest to the final exam, because it is examined only there.
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