Exercises on Probability

1) Arithmetic mean

a) Demonstrate, that the sum of the deviations of set of \(N\) numbers \(x_i\) from their arithmetic mean \(\bar{x}\) is equals to zero. i.e. $\(\sum_{i=1}^{N} \varepsilon_i = \sum_{i=1}^{N}(x-\bar{x}) = ... = 0\)$

b) Is this property also true for the continuous case? If so demonstrate it.
Hint: the sum of the deviations will now be represented by an integral:
\(\tau=\int_{-\infty}^{\infty}(x-\bar{x})f(x)dx\) where the function \(f(x)\) has this property \(\int_{-\infty}^{\infty}(x-\bar{x})f(x)dx=1\)

2) Tossing coins

A fair coin is tossed eight times. Let the random variable be the number of heads that appear.

2.1) What is the probability that exactly 4 heads will appear?

2.2) What is the probability that you get either all heads or all tails?