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2.3 Probability & Statistics

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FE Electrical and Computer Exam Exam

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1 FE Electrical & Computer: Exam Map, Handbook & Strategy

1.1 Current Exam Facts 1.2 CBT Format & Scaled Scoring 1.3 The FE Reference Handbook 1.4 Approved Calculators & Test-Day Logistics 1.5 Study Plan & Time Strategy

2 Mathematics, Economics & Ethics

2.1 Algebra, Calculus & Differential Equations 2.2 Complex Numbers & Laplace Transforms 2.3 Probability & Statistics 2.4 Engineering Economics 2.5 Ethics & Professional Practice

3 Circuits, Electronics & Electromagnetics

3.1 DC Circuit Analysis 3.2 AC Steady-State & Phasors 3.3 Transient Response (RL, RC, RLC) 3.4 Electronics & Devices 3.5 Electromagnetics

4 Signals, Systems, Controls & Communications

4.1 Linear Systems & Transfer Functions 4.2 Signal Processing & Sampling 4.3 Control Systems 4.4 Communications

5 Digital Systems, Computers, Networks & Software

5.1 Number Systems & Boolean Algebra 5.2 Combinational & Sequential Logic 5.3 Computer Architecture & Microprocessors 5.4 Computer Networks 5.5 Software Engineering & Algorithms

6 Power, Materials & Final Readiness

6.1 Power Fundamentals & Three-Phase 6.2 Transformers & Rotating Machines 6.3 Power Electronics & Transmission 6.4 Properties of Electrical Materials & Engineering Sciences 6.5 Integrated Practice & Final Readiness

2.3 Probability & Statistics

Key Takeaways Explain This Topic AI

Probability and Statistics covers distributions, expected value, dispersion measures, regression, hypothesis testing, and reliability on the FE Electrical and Computer exam.

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.

Last updated: June 2026

What the exam asks

Probability and Statistics 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): μ = (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: σ = 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).

Worked example (mean, median, std dev): For the data set {2, 4, 4, 6, 9}, the mean is μ = (2+4+4+6+9)/5 = 25/5 = 5. The median (middle of the sorted list) is 4 and the mode is 4. The squared deviations from the mean are 9, 1, 1, 1, 16, summing to 28. The population standard deviation is σ = sqrt(28/5) = sqrt(5.6) = 2.37; the sample standard deviation is s = sqrt(28/4) = sqrt(7) = 2.65. The denominator choice changes the answer, which is exactly the distractor the FE exploits.

Permutations and combinations

When order matters, use permutations P(n,r) = n!/(n-r)!. When order does not, use combinations C(n,r) = n!/[r!(n-r)!]. For example, the number of 3-member committees from 5 people is C(5,3) = 5!/(3!2!) = 10, while the number of ordered finishes for 3 of 5 racers is P(5,3) = 5!/2! = 60. The single most common FE counting error is choosing permutations when the problem does not care about order (or vice versa) — read for the words "arrangement/order" versus "group/committee/selection."

Basic probability rules

For mutually exclusive events, P(A or B) = P(A) + P(B); in general P(A or B) = P(A) + P(B) - P(A and B). For independent events, P(A and B) = P(A)·P(B). Conditional probability is P(A|B) = P(A and B)/P(B). These rules combine with counting: the probability of drawing 2 defective parts from a bin is the count of favorable combinations over the count of total combinations. Keep probabilities between 0 and 1 — a result above 1 means you added when you should have multiplied.

  • Expected value, distributions, regression, and reliability
  • 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), and the variance is Var(X) = E[X²] - (E[X])². This shortcut (mean of the squares minus the square of the mean) is faster than the deviation form under exam timing.

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 λ; mean = var = λ
  • Normal (Gaussian) Continuous measurement error, noise μ, σ; bell curve
  • Exponential Time between events, component lifetime λ; mean = 1/λ

The binomial probability of exactly k successes is P(k) = C(n,k) p^k (1-p)^(n-k). For the normal distribution , standardize with the z-score z = (x - μ)/σ, then read the standard normal table in the Handbook. The empirical rule says about 68% of values lie within ±1σ, 95% within ±2σ, and 99.7% within ±3σ.

Worked example (binomial): A circuit has a 0.1 chance of a defective resistor. In a batch of 5, what is the probability of exactly 1 defect? P(1) = C(5,1)(0.1)¹(0.9)⁴ = 5(0.1)(0.6561) = 0.328, about 33%.

Linear regression and hypothesis testing

Least-squares fits y = a + bx with slope b = S_xy/S_xx; the line passes through the centroid (x̄, ȳ). The correlation coefficient r ranges from -1 to +1; r² is the fraction of variance explained. In hypothesis testing you state a null H0 and alternative H1, choose significance level α (often 0.05), and reject H0 if the test statistic falls in the rejection region. A Type I error rejects a true null (probability α); a Type II error fails to reject a false null (probability β).

Reliability

For n components in series , all must 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 λ, reliability R(t) = e^(-λt) and the mean time to failure (MTTF) = 1/λ.

Worked example (reliability and MTTF): A power-supply component has a constant failure rate λ = 0.001 per hour. Its MTTF is 1/λ = 1,000 hours, and the probability it survives 500 hours is R(500) = e^(-0.001·500) = e^(-0.5) = 0.607, about 61%. Adding a second identical unit in parallel raises survival to 1 - (1 - 0.607)² = 1 - 0.154 = 0.846, about 85% — redundancy buys reliability at the cost of a duplicated part. Worked example (z-score): If exam scores are normal with μ = 70 and σ = 8, a score of 86 has z = (86 - 70)/8 = 2.0, so by the empirical rule only about 2.5% of scores exceed it (half of the 5% lying beyond ±2σ).