Fundamentals of Engineering Exam (FE) Mathematics, Probability and Statistics Flashcards

24 sample cards from the Mathematics, Probability and Statistics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State the sum of a finite geometric series and the condition for the infinite sum to converge.

   $S_{n} = a_{1}\dfrac{1-r^{n}}{1-r}$; the infinite series converges to $\dfrac{a_{1}}{1-r}$ when $|r| < 1$.

2. Write the Taylor series expansion of $f(x)$ about $x=a$.

   $f(x) = \displaystyle\sum_{n=0}^{\infty} \dfrac{f^{(n)}(a)}{n!}(x-a)^{n}$

3. What distinguishes a first-order linear ODE, and what is its integrating factor?

   It has the form $\dfrac{dy}{dx} + P(x)y = Q(x)$; the integrating factor is $\mu(x) = e^{\int P(x)\,dx}$.

4. For the second-order linear homogeneous ODE $ay''+by'+cy=0$, what is the characteristic equation?

   $ar^{2} + br + c = 0$; its roots determine the form of the general solution.

5. For $ay''+by'+cy=0$ with distinct real roots $r_{1}, r_{2}$, what is the general solution?

   $y = C_{1}e^{r_{1}x} + C_{2}e^{r_{2}x}$

6. How do you separate variables to solve $\dfrac{dy}{dx} = g(x)h(y)$?

   Rewrite as $\dfrac{dy}{h(y)} = g(x)\,dx$, then integrate both sides: $\displaystyle\int \dfrac{dy}{h(y)} = \int g(x)\,dx + C$.

7. State the addition rule of probability for two events $A$ and $B$.

   $P(A \cup B) = P(A) + P(B) - P(A \cap B)$; for mutually exclusive events the last term is $0$.

8. Define conditional probability and the multiplication rule.

   $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$, so $P(A \cap B) = P(A \mid B)\,P(B)$. If independent, $P(A\cap B)=P(A)P(B)$.

9. State Bayes' theorem.

   $P(A \mid B) = \dfrac{P(B \mid A)\,P(A)}{P(B)}$

10. Give the formulas for permutations and combinations of $n$ items taken $r$ at a time.

    Permutations: $P(n,r) = \dfrac{n!}{(n-r)!}$. Combinations: $C(n,r) = \dfrac{n!}{r!\,(n-r)!}$.

11. State the probability mass function, mean, and variance of the binomial distribution.

    $P(X=k) = \binom{n}{k}p^{k}(1-p)^{n-k}$, with mean $\mu = np$ and variance $\sigma^{2} = np(1-p)$.

12. State the mean and variance of a Poisson distribution with parameter $\lambda$.

    Both equal $\lambda$: $\mu = \lambda$ and $\sigma^{2} = \lambda$, with $P(X=k)=\dfrac{\lambda^{k}e^{-\lambda}}{k!}$.

13. How is a value $x$ converted to a standard normal z-score?

    $z = \dfrac{x - \mu}{\sigma}$

14. Define sample mean and sample variance for data $x_{1},\dots,x_{n}$.

    $\bar{x} = \dfrac{1}{n}\displaystyle\sum_{i=1}^{n} x_{i}$ and $s^{2} = \dfrac{1}{n-1}\displaystyle\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}$.

15. Compare the mean, median, and mode as measures of central tendency.

    The mean is the arithmetic average (sensitive to outliers); the median is the middle value (robust to outliers); the mode is the most frequent value.

16. What does the Central Limit Theorem state about the sampling distribution of the sample mean?

    For large $n$, the sample mean $\bar{x}$ is approximately normally distributed with mean $\mu$ and standard error $\dfrac{\sigma}{\sqrt{n}}$, regardless of the population's distribution.

17. Write the general form of a confidence interval for a population mean (known $\sigma$).

    $\bar{x} \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}$

18. In hypothesis testing, what are Type I and Type II errors?

    A Type I error rejects a true null hypothesis (probability $\alpha$); a Type II error fails to reject a false null hypothesis (probability $\beta$).

19. State the Newton-Raphson iteration formula for root finding.

    $x_{n+1} = x_{n} - \dfrac{f(x_{n})}{f'(x_{n})}$

20. Describe the bisection method for finding a root.

    Given $[a,b]$ with $f(a)f(b)<0$, repeatedly evaluate the midpoint $c=\frac{a+b}{2}$ and replace whichever endpoint keeps the sign change bracketing the root, halving the interval each step.

21. State the trapezoidal rule for approximating $\displaystyle\int_{a}^{b} f(x)\,dx$ with step $h$.

    $\displaystyle\int_{a}^{b} f(x)\,dx \approx \dfrac{h}{2}\left[f(x_{0}) + 2\sum_{i=1}^{n-1}f(x_{i}) + f(x_{n})\right]$

22. State Simpson's $\frac{1}{3}$ rule for numerical integration (requires an even number of intervals).

    $\displaystyle\int_{a}^{b} f(x)\,dx \approx \dfrac{h}{3}\left[f(x_{0}) + 4\sum_{\text{odd}}f(x_{i}) + 2\sum_{\text{even}}f(x_{i}) + f(x_{n})\right]$

23. State Euler's method for numerically solving $y' = f(x,y)$ with step size $h$.

    $y_{n+1} = y_{n} + h\,f(x_{n}, y_{n})$

24. Compare Euler's method and the fourth-order Runge-Kutta (RK4) method for solving ODEs.

    Euler's method is first-order accurate (local error $\sim h^{2}$) using one slope evaluation per step; RK4 is fourth-order accurate using a weighted average of four slope evaluations per step, giving far greater accuracy for the same step size.

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