Problem Set 1 - Preliminaries

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Problem Set 1 - Preliminaries#

(Check Canvas for due dates and office hours)

  1. Answer (Hint: use \(P(\Omega) = 1\))

    • 1.1 \(g(\omega) = 1 \,\, \forall \,\, \omega \in \Omega; \Omega = [0,1]\). Is the function \(g\) a pdf, a pmf, or neither?

    • 1.2. \(g(\omega) = 1/2 \,\, \forall \,\, \omega \in \Omega; \Omega = \{Stone, Paper, Scissor\} \). Is the function \(g\) a pdf, a pmf, or neither?

    • 1.3. \(g(\omega) = |\omega| \,\, \forall \,\, \omega \in \Omega; \Omega = [-1,1] \). Is the function \(g\) a pdf, a pmf, or neither? Note: \(|\omega| = \begin{cases} \omega, \,\, if \,\, \omega \geq 0 \\ -\omega, \,\, if \,\, \omega < 0 \\ \end{cases} \) (known as absolute value function)

    • 1.4. \(g(\omega) = \frac{1}{\sqrt{8\pi}}\exp(-\frac{\omega^2}{8}) \forall \,\, \omega \in \Omega; \Omega = (-\infty,\infty)\). Is the function \(g\) a pdf, a pmf, or neither?

  2. For a uniform distribution \(U(5,8)\)

    • What is the probability \(p(5.5)\)?

    • What is \(f(5.5)\)?

    • What are \(F(0.2)\), \(F(5.5)\), and \(F(8.5)\)?

  3. Shade to show appropriate values on a figure of normal pdf with mean \(\mu\) and variance \(\sigma^2\). Draw a figure for each case.

    • \(p(x \leq a)\), where \(a < \mu\).

    • \(p(x = b)\), where \(b < \mu\).

    • \(F(c)\), where \(c > \mu\).

    • \(F(d)\), where \(d = \mu\).

  4. From problem 3, provide the corresponding values for \(\mu=0\) and \(\sigma^2 = 1\)

    • \(a = -0.2\).

    • \(b = -0.1\).

    • \(c = 0.25\).

    • \(d = \mu = 0\).

    • Additionally, provide the values of \(\Phi(0.25)\) and \(\Phi(0)\). Comment on their similarity/difference from \(F(0.25)\) and \(F(0)\).

  5. For a Binomial distribution \(B(10,0.3)\),

    • Find \(p(8)\).

    • Find \(p(x \leq 8)\).

    • Find \(F(1)\)

  6. Textbook Problem 3.25: A mechatronic assembly is subjected to a final functional test. Suppose that defects occur at random in these assemblies, and that defects occur according to a Poisson distribution with parameter \(\lambda= 0.02\).

    • What is the probability that an assembly will have exactly one defect?

    • What is the probability that an assembly will have one or more defects?

    • Suppose that you improve the process so that the occurrence rate of defects is cut in half to \(\lambda= 0.01\). What effect does this have on the probability that an assembly will have one or more defects?

  7. Textbook Problem 3.38: Surface-finish defects in a small electric appliance occur at random with a mean rate of 0.1 defects per unit. Find the probability that a randomly selected unit will contain at least one surface-finish defect.

  8. Textbook Problem 3.48: A lightbulb has a normally distributed light output with a mean of 5,000 end foot-candles and a standard deviation of 50 end foot-candles. Find a lower specification limit such that only 0.5% of the bulbs will not exceed this limit.

  9. Write five possible samples (a.k.a. generate your own random numbers) each of size \(n=6\) from

    • \(\mathcal{N}(1.0,1.0)\)

    • \(B(5,0.2)\)

    • \(U(5,6)\)

  10. From the previous answer, make a single sample of \(n=30\) from the five samples of \(\mathcal{N}(1.0,1.0)\). Draw corresponding histogram of the combined sample (by hand).

Acknowledgements#

  • Congguang (Steven) Chen - Fall 2025

  • Sunshine McDonald - Fall 2025

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