The store has three identical automatic checkouts which, during a day, each triggered 20 checks. We denote $X_1, X_2$ and $X_3$ the random variables associating to each checkout the number of errors detected during this day. We admit that the random variables $X_1, X_2$ and $X_3$ are independent of each other and each follow a binomial distribution $\mathscr{B}(20; 0.165)$.
Determine the exact values of the expectation and variance of the random variable $X_1$.
We define the random variable $S$ by $S = X_1 + X_2 + X_3$. Justify that $E(S) = 9.9$ and that $V(S) = 8.2665$. For this question, we will use 10 as the value of $E(S)$. Using the Bienaymé-Chebyshev inequality, show that the probability that the total number of errors on the day is strictly between 6 and 14 is greater than 0.48.
The store has three identical automatic checkouts which, during a day, each triggered 20 checks. We denote $X_1, X_2$ and $X_3$ the random variables associating to each checkout the number of errors detected during this day.\\
We admit that the random variables $X_1, X_2$ and $X_3$ are independent of each other and each follow a binomial distribution $\mathscr{B}(20; 0.165)$.
\begin{enumerate}
\item Determine the exact values of the expectation and variance of the random variable $X_1$.
\item We define the random variable $S$ by $S = X_1 + X_2 + X_3$.
Justify that $E(S) = 9.9$ and that $V(S) = 8.2665$.\\
For this question, we will use 10 as the value of $E(S)$.\\
Using the Bienaymé-Chebyshev inequality, show that the probability that the total number of errors on the day is strictly between 6 and 14 is greater than 0.48.
\end{enumerate}