text: # Study Session 2: Probability and Statistics
  
  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 $$\sigma_x$$