Welcome to the notes for Statistical Inference. The course is part of the MSc in Statistics for Data Science from Carlos III University of Madrid.

The course is designed to have, roughly, one session per each main topic in the syllabus. The schedule is tight due to time constraints, which will inevitably make the exposition of certain methods somehow superficial. Nevertheless, the course and exercises will hopefully give you a respectable panoramic view of the fundamentals of (mostly univariate) statistical inference. A broad view of the syllabus and its planning is:

  1. Preliminaries (first session)
  2. Introduction to statistical inference (second session)
  3. Point estimation (third session)
  4. Estimation methods (fourth session)
  5. Confidence intervals (fifth session)
  6. Hypothesis tests (sixth session)

Some logistics for the development of the course follow:

  • Office hours are described in the Aula Global (right panel).
  • Questions and comments during lectures are most welcome. Particularly if these are clarifications, comments, or alternative perspectives that may help the rest of the class. So just go ahead and fire!
  • Detailed course evaluation guidelines can be found in the Aula Global. Recall that participation in lessons is positively evaluated.


These notes are possible due to the existence of the incredible pieces of software by Xie (2016), Xie (2020), Allaire et al. (2020), Xie and Allaire (2020), and R Core Team (2020). Also, certain hacks to improve the design layout have been possible due to the outstanding work by Úcar (2018). The icons used in the notes were designed by madebyoliver, freepik, and roundicons from Flaticon.

More general and complete monographs on statistical inference include the books by Casella and Berger (2002), Silvey (1975), Lehmann and Casella (1998), Lehmann and Romano (2005), and Shao (2003).

Last but not least, the notes have benefited from contributions from the following people:


Contributions, reporting of typos, and feedback on the notes are very welcome. Just send an email to and give me a good reason for writing your name in the list of contributors!


All the material in these notes is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Public License (CC BY-NC-ND 4.0). You may not use this material except in compliance with the aforementioned license. The human-readable summary of the license states that:

  • You are free to:
    • Share – Copy and redistribute the material in any medium or format.
  • Under the following terms:
    • Attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
    • NonCommercial – You may not use the material for commercial purposes.
    • NoDerivatives – If you remix, transform, or build upon the material, you may not distribute the modified material.


You may use the following \(\mathrm{B{\scriptstyle{IB}} \! T\!_{\displaystyle E} \! X}\) entry when citing these notes:

    title        = {A First Course on Statistical Inference},
    author       = {Molina Peralta, I. and Garc\'ia-Portugu\'es, E.},
    year         = {2024},
    note         = {Version 2.4.1. ISBN 978-84-09-29680-4},
    url          = {}

You may also want to use the following template:

Molina Peralta, I. and García-Portugués, E. (2024). A First Course on Statistical Inference. Version 2.4.1. ISBN 978-84-09-29680-4. Available at


Allaire, J. J., Y. Xie, J. McPherson, J. Luraschi, K. Ushey, A. Atkins, H. Wickham, J. Cheng, W. Chang, and R. Iannone. 2020. rmarkdown: Dynamic Documents for R.
Casella, G., and R. L. Berger. 2002. Statistical Inference. Duxbury Advanced Series. Duxbury-Thomson Learning.
Lehmann, E. L., and G. Casella. 1998. Theory of Point Estimation. Second. Springer Texts in Statistics. New York: Springer-Verlag.
Lehmann, E. L., and J. P. Romano. 2005. Testing Statistical Hypotheses. Third. Springer Texts in Statistics. New York: Springer.
R Core Team. 2020. R: A Language and Environment for Statistical Computing. Vienna.
Shao, J. 2003. Mathematical Statistics. Second. Springer Texts in Statistics. New York: Springer-Verlag.
Silvey, S. D. 1975. Statistical Inference. Monographs on Statistical Subjects. New York: Chapman; Hall.
Úcar, I. 2018. “Energy Efficiency in Wireless Communications for Mobile User Devices.” PhD thesis, Universidad Carlos III de Madrid.
Xie, Y. 2016. Bookdown: Authoring Books and Technical Documents with R Markdown. The r Series. Boca Raton: CRC Press.
———. 2020. knitr: A General-Purpose Package for Dynamic Report Generation in R.
Xie, Y., and J. J. Allaire. 2020. tufte: Tufte’s Styles for R Markdown Documents.