2.3 Probability & Statistics
Key Takeaways
- Probability and Statistics is a focused 4-6 of 110 questions covering distributions, expected value, dispersion measures, regression, hypothesis testing, and reliability.
- Expected value is the probability-weighted sum E[X] = sum(x_i p_i); variance is E[X^2] - (E[X])^2 and standard deviation is its square root.
- Sample variance divides by (n - 1); population variance divides by N. Picking the wrong denominator is a classic FE trap.
- For a normal distribution, about 68%, 95%, and 99.7% of values fall within one, two, and three standard deviations of the mean (the empirical rule).
- Reliability of series components multiplies (R = R1 x R2 x ...); exponential-lifetime reliability is R(t) = e^(-lambda t), where 1/lambda is the mean time to failure.
What the exam asks
Probability and Statistics is 4-6 of 110 questions, but the questions are formulaic and fast points if you know the right expression. The Handbook supplies the formulas; you supply recognition and clean arithmetic.
Central tendency and dispersion
- Mean (arithmetic average): mu = (sum of x_i) / n.
- Median: the middle value when sorted; robust to outliers.
- Mode: the most frequent value.
- Variance: spread about the mean.
- Standard deviation: sigma = sqrt(variance), in the same units as the data.
Watch the denominator. Population variance divides by N; sample variance divides by (n - 1) (Bessel's correction). The FE often supplies a small sample and expects (n - 1).
Expected value and variance of a random variable
For a discrete random variable, the expected value is the probability-weighted sum:
E[X] = sum(x_i · p_i)
The variance is:
Var(X) = E[X^2] - (E[X])^2
This shortcut (mean of the squares minus the square of the mean) is faster than the deviation form under exam timing.
Distributions, regression, and testing
Common distributions
| Distribution | Models | Key parameter(s) |
|---|---|---|
| Binomial | Count of successes in n independent trials | n, p; mean = np, var = np(1-p) |
| Poisson | Counts of rare events per interval | lambda; mean = var = lambda |
| Normal (Gaussian) | Continuous measurement error, noise | mu, sigma; bell curve |
| Exponential | Time between events, component lifetime | lambda; mean = 1/lambda |
For the normal distribution, standardize with the z-score z = (x - mu) / sigma, then read the standard normal table in the Handbook. The empirical rule says about 68% of values lie within +/-1 sigma, 95% within +/-2 sigma, and 99.7% within +/-3 sigma.
Linear regression
Least-squares fits y = a + bx by minimizing squared residuals. The slope is b = S_xy / S_xx and the line passes through the centroid (x-bar, y-bar). The correlation coefficient r ranges from -1 to +1; r^2 is the fraction of variance explained.
Hypothesis testing (concept level)
State a null hypothesis (H0) and alternative (H1), choose a significance level alpha (commonly 0.05), compute a test statistic, and reject H0 if it falls in the rejection region. A Type I error rejects a true null (probability alpha); a Type II error fails to reject a false null (probability beta).
Reliability
For n components in series, the system works only if all work: R_sys = R1 × R2 × ... × Rn. For parallel redundancy, the system fails only if all fail: R_sys = 1 - (1-R1)(1-R2)...(1-Rn). With a constant failure rate lambda, reliability R(t) = e^(-lambda t), and the mean time to failure (MTTF) = 1/lambda.
A discrete random variable takes values 0, 1, 2 with probabilities 0.2, 0.5, 0.3. What is its expected value?
Three independent components in series have reliabilities 0.9, 0.95, and 0.98. What is the system reliability?