# Two-dimensional Random Variables¶

(c) Tobias Hossfeld (Aug 2021)

This script and the figures are part of the following book. The book is to be cited whenever the script is used (copyright CC BY-SA 4.0):
Tran-Gia, P. & Hossfeld, T. (2021). Performance Modeling and Analysis of Communication Networks - A Lecture Note. Würzburg University Press. https://doi.org/10.25972/WUP-978-3-95826-153-2

Penalty shootout We are looking at a penalty shootout between two teams: FC Bayern München and VFL Bochum. Every team has to shoot five penalties. FCB scores with a probability of 95%, while VFL scores with a probability of 85%. What is the joint probability that FCB scores $i$-times and VFL scores $j$-times. What is the probability that VFL scores more often than FCB?

## Joint Probability Mass Function (PMF)¶

We are considering the joint probability mass function $P(X_1=i,X_2=j)$ that FC Bayern scores $i$ goals and VFL scores $j$ goals. The probabilities are independent of each other.

$P(X_1=i,X_2=j) = P(X_1=i)\cdot P(X_2=j)$

What is the probability that VFL Bochum wins: $P(X_1>X_2)$?