# Study Session 2: Probability and Statistics

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The uncertain nature of things in the quantum world make it necessary for us to use probability and statistics to describe the likelihood of finding a system in a certain state. That means a short review is useful.

Flip a coin 20 times and record the outcome. How many times did your coin flip result in heads? How often do you expect to get heads?

If you roll a six-sided die 60 times, how many times do you expect your roll to result in? Why?

If \(N_j\) is the number of times that you measure outcome \(j\), out of \(N\) total repetitions, then

\(p_j=\lim_{N\rightarrow \infty}\frac{N_j}{N}\)

What values can \(p_j\) have? What does it mean if \(p_j=1\)? what deos it mean if \(p_j=0\)?

If \(N_j\) has \(n\) possible values, what is the value of \(\sum_{j=1}^{n}N_j\)?

What is the value of \(\sum_{j=1}^{n}p_j\)?

Using \(p_j\), how would you describe the average value of your die roll?

We can write the average or mean of a measurement using

\(\left \langle x \right \rangle =\sum_{j=1}^{n}x_jp_j=\sum_{j=1}^{n}x_jp(x_j)\)

What is the probability of rolling any particular value 1-6 on a die? What is the average value that you will roll?

In addition to the mean or average, we can compute the second moment, \( \left \langle x^2 \right \rangle = \sum_{j=1}^{n}x_j^2p_j\) and with this, we define the variance

\(\sigma_x^2=\left \langle (x-\left \langle x \right \rangle )^2 \right \rangle = \sum_{j=1}^{n}(x-\left \langle x \right \rangle )^2p_j\)

How do you get a large value for the variance? How do you get a small value for the variance? What does the variance measure?

Complete the square in the sum for the variance and simplify the expression.

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What happens when our distribution is continuous instead of discrete? Then the probability of finding the system in an infinitesimal position between \(x\) and \(x+dx\) is \(p(x)dx\).

How do we express the probability of finding a particle between \(x=a\) and \(x=b\)?

The most common distribution we'll use is a Gaussian, \(p(x)dx=ce^{-x^2/2a^2}dx\) for \(-\infty < x < \infty \).

Find \(c, \left \langle x \right \rangle , \sigma_x^2\), and \(\sigma_x\)