Statistics and Risk Analysis: free practice, theory and problems

1. Probability

What is it about?

Probability theory gives you tools to handle uncertainty: how likely is it that a component fails, that a test gives the correct result, or that two events happen at the same time? As an engineer you use this for reliability calculations, quality control and risk assessment.

Concepts and formulas

• Probability $P(A)$ lies between 0 and 1. $P(A^c)=1-P(A)$ (complement).
• Union: $P(A\cup B) = P(A)+P(B)-P(A\cap B)$. For disjoint (mutually exclusive) events $P(A\cap B)=0$, so $P(A\cup B)=P(A)+P(B)$.
• Conditional probability: $P(A\mid B) = \dfrac{P(A\cap B)}{P(B)}$ – the probability of $A$ given that $B$ has occurred.
• Independence: $A$ and $B$ are independent if $P(A\cap B)=P(A)P(B)$. Disjoint events (with positive probability) are actually dependent, since one rules out the other.
• Bayes' formula: $P(A\mid B) = \dfrac{P(B\mid A)P(A)}{P(B)}$ – used to flip the conditioning (e.g. from "test result given condition" to "condition given test result").
• Reliability: components in series require all of them to work: $R=R_1R_2\cdots$. Components in parallel require at least one to work: $R=1-(1-R_1)(1-R_2)\cdots$.
• Combinatorics: number of ways to choose $k$ out of $n$ regardless of order: $\binom{n}{k}=\dfrac{n!}{k!(n-k)!}$. With order taken into account (permutations of $k$ out of $n$): $n(n-1)\cdots(n-k+1)$.

How to solve the problems

1. Identify what is being asked: union, intersection, complement, conditional probability or Bayes.
2. Check whether the events are independent, disjoint or neither – that decides which formula applies.
3. For reliability: sketch whether the components are in series, parallel or a combination.
4. For counting: decide whether the order matters (permutation) or not (combination).

Example

A fire alarm has two independent sensors in parallel, each with reliability 0.95 (probability of working when there is a fire). What is the probability that the alarm does NOT trigger? The system works if at least one sensor works: $R = 1-(1-0.95)(1-0.95) = 1-0.05^2 = 1-0.0025 = 0.9975$. The probability that the alarm does not trigger is the complement: $1-0.9975 = 0.0025$.

Common mistakes

• Believing that disjoint events are independent. They are actually strongly dependent: if one happens, the other cannot.
• Adding probabilities without subtracting the intersection when the events are not disjoint.
• Confusing series and parallel: series is weaker (everything must work), parallel is stronger (one is enough).
• Using combinations when order actually matters, or the other way around.

Concepts in this part

Practise probability in the app →

2. Distributions

What is it about?

Data from measurements or processes vary. Probability distributions give a mathematical model of this variation, so you can compute probabilities, expected values and how reliable a measurement is. Measures of center (mean, median, mode) and spread (variance, standard deviation) summarize data; distributions such as the normal, binomial and Poisson distributions describe how random outcomes arise.

Concepts and formulas

• Mean $\bar x = \tfrac1n\sum x_i$. Median: the middle value in sorted order. Mode: the value that occurs most often.
• Variance and standard deviation: $\sigma^2=\text{Var}(X)$, $\sigma=\sqrt{\text{Var}(X)}$. Sample variance uses $n-1$ in the denominator: $s^2=\tfrac{1}{n-1}\sum(x_i-\bar x)^2$ – this gives an unbiased estimator of the population variance (Bessel's correction).
• Normal distribution: symmetric bell shape. The 68–95–99.7 rule: about 68%, 95% and 99.7% of the probability lies within $\mu\pm1\sigma$, $\pm2\sigma$ and $\pm3\sigma$. z-score: $z=(x-\mu)/\sigma$.
• Binomial distribution $X\sim\text{Bin}(n,p)$: the number of successes in $n$ independent trials. $E[X]=np$, $\text{Var}(X)=np(1-p)$, $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.
• Poisson distribution: the number of events in a time interval with rate $\lambda$: $P(X=k)=\lambda^k e^{-\lambda}/k!$.
• Exponential distribution: waiting time or lifetime with a constant rate: $P(T>t)=e^{-t/\mu}$.
• Central limit theorem (CLT): the mean of many independent measurements becomes approximately normally distributed, regardless of the original distribution.
• Correlation $r\in[-1,1]$: the strength of the linear relationship between two variables. Correlation is not the same as causation.

How to solve the problems

1. Decide which kind of distribution the situation describes (number of successes → binomial, events per time → Poisson, continuous measurement → normal).
2. Find the parameters ($n,p$ or $\lambda$ or $\mu,\sigma$).
3. Use the correct formula for what is asked: expectation, variance, the probability of a specific value, or the z-score.
4. For the normal distribution: convert to a z-score before applying a probability rule.

Example

A machine produces parts where 8% are defective. You inspect 5 random parts. What is the probability of exactly 1 defective part? This is binomial with $n=5$, $p=0.08$ and $k=1$: $P(X=1)=\binom{5}{1}(0.08)^1(0.92)^4 = 5\cdot 0.08\cdot 0.7164 \approx 0.2866$.

Common mistakes

• Using $n$ instead of $n-1$ in the sample variance (or the other way around for an entire population).
• Believing the mode is always unique – a data set can have several modes or none.
• Applying the normal distribution to data that is clearly not symmetric (e.g. lifetimes, which are often exponentially or Weibull distributed).
• Interpreting a high correlation as proof of causation.

Concepts in this part

Practise distributions in the app →

3. Inference and risk

What is it about?

Inference is about drawing conclusions about an entire population from a sample – for example whether a new process really gives better quality, or how confident you can be in a mean value. Risk analysis uses probability and consequence to prioritize which hazards are worth acting on. Both are about making decisions under uncertainty, backed by numbers instead of gut feeling.

Concepts and formulas

• Hypothesis testing: $H_0$ (the null hypothesis, "no effect") is tested against $H_1$. The p-value is the probability of a result at least as extreme, given that $H_0$ is true. Reject $H_0$ if the p-value is smaller than the significance level $\alpha$ (often 0.05).
• One-tailed vs. two-tailed test: a two-tailed test checks deviations in both directions from $H_0$; a one-tailed test checks only one direction (e.g. "is the new method better", not "different").
• Type I error: rejecting a true $H_0$ (probability $\alpha$). Type II error: retaining a false $H_0$ (probability $\beta$).
• Standard error: $\sigma_{\bar x}=\sigma/\sqrt n$. Confidence interval for $\mu$ with known $\sigma$: $\bar x \pm z\cdot\sigma/\sqrt n$ ($z=1.96$ for 95%). If $\sigma$ is unknown, the $t$-distribution is used with the sample's standard deviation instead – necessary especially for small samples.
• Linear regression: the slope $b=\dfrac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(x_i-\bar x)^2}$. $R^2$ is the fraction of the variation in $y$ that the model explains (between 0 and 1).
• Risk = probability × consequence. FMEA (Failure Mode and Effects Analysis): a systematic review of possible failure modes, causes and consequences. ALARP: risk should be reduced as far as reasonably practicable.
• Reliability: $A=\dfrac{MTBF}{MTBF+MTTR}$ (availability), $\lambda=1/MTBF$ (failure rate).

How to solve the problems

1. Identify what is being tested ($H_0$/$H_1$) or which risk measure is being asked for.
2. Find the necessary quantities: $\bar x$, $\sigma$ (or $s$), $n$, and possibly probability and consequence.
3. Plug the numbers into the correct formula, and watch the units.
4. Interpret the answer in context: is the p-value below $\alpha$? Is the risk acceptable?

Example

You measure $x=2, 4, 6, 8$ (hours) and $y=5, 7, 8, 14$ (tonnes produced). Find the slope of the regression line. $\bar x=5$ and $\bar y=8.5$. The deviations in $x$: $-3,-1,1,3$; in $y$: $-3.5,-1.5,-0.5,5.5$. The products sum to $28$, and $\sum(x_i-\bar x)^2=20$. The slope is $b=28/20=1.4$.

Common mistakes

• Believing the p-value is the probability that $H_0$ is true. It is the probability of the data, given that $H_0$ is true.
• Using $z$ instead of $t$ when $\sigma$ is unknown and the sample is small.
• Believing a confidence interval gets narrower with a higher confidence level – the opposite is true.
• Ranking risk by only probability or only consequence, instead of the product of the two.

Concepts in this part

Practise inference and risk in the app →

4. Regression and correlation

What is it about?

You have measured two quantities, for example temperature and power consumption, and wonder whether they are related. Regression finds the straight line that fits the points best, so you can predict new values. Correlation is a single number that tells you how closely the points follow a straight line.

Concepts and formulas

• The regression line $\hat y = a + b x$ is the line with the smallest possible sum of squared deviations (least squares).
• The slope and the intercept are computed from the means $\bar x$ and $\bar y$:

• The correlation coefficient $r$ lies between $-1$ and $1$. Close to $1$: strong positive relationship. Close to $-1$: strong negative. Close to $0$: no linear relationship.

• Coefficient of determination $r^2$: the share of the variation in $y$ explained by the line. $r = 0.9$ gives $r^2 = 0.81$, i.e. 81 %.
• Residual: measured value minus the value from the line, $e_i = y_i - \hat y_i$.

How to solve the problems

1. Compute $\bar x$ and $\bar y$.
2. Make a table with $x_i - \bar x$, $y_i - \bar y$, their product and the square.
3. Add up the columns and compute $b$, then $a = \bar y - b\bar x$.
4. Insert $x$ in $\hat y = a + bx$ to predict.

Example

The points $(1, 2)$, $(2, 4)$, $(3, 5)$ and $(4, 7)$.

1. $\bar x = 2.5$ and $\bar y = 4.5$.
2. $\sum (x_i-\bar x)(y_i-\bar y) = 3.75 + 0.25 + 0.25 + 3.75 = 8$ and $\sum (x_i-\bar x)^2 = 5$.
3. $b = 8/5 = 1.6$ and $a = 4.5 - 1.6\cdot 2.5 = 0.5$.
4. The line is $\hat y = 0.5 + 1.6x$. For $x = 5$ it predicts $\hat y = 8.5$.

Common mistakes

• Believing that correlation means cause. Ice cream sales and drowning accidents are related because both increase in summer.
• Using the line far outside the measurements (extrapolation). The relationship need not hold there.
• Swapping $x$ and $y$. The line for $y$ given $x$ is not the same as for $x$ given $y$.
• Reading $r = 0$ as "no relationship". There may be a strong but curved relationship.

Concepts in this part

Practise regression and correlation in the app →