# Questions

• Why should we quote error bars on our measurements?
• What distinguishes a sample mean from a population mean?
• What integrals should be calculated to give the expectation and variance?
• State the central limit theorem and explain what each of the terms in this theorem represent.
• How is the cumulative probability distribution function defined and how is the probability density function defined?
• What is the variance of the random variable $S = \frac{1}{N} \sum_{i=1}^N X_i$ if each of the $X_i$s in this sum is a independent and identically distributed random variable taken from a distribution with variance $\sigma$?
• Explain in your own words why a sample mean gives an estimate of the true population mean.
• Give the expression that allows you to calculate an estimate of the true population from a sample taken from that population and explain how this result is derived.
• We can calculate a weighted sample average as $\mu_S = \frac{1}{W} \sum_{i=1}^{N_S} w_i x_i$, where $W= \sum_{i=1}^{N_S} w_i$. In these expressions the $x_i$s are the values of the quantities in our samples and the $w_i$s are the sample weights. Show, using the maths from the video, that an appropriate estimator for the popupation variance for this weighted sample is $\sigma^2 = \frac{W}{W - \frac{W_2}{W} } \frac{1}{W} \sum_{i=1}^{N_S} w_i (x_i - \mu_S)^2$, where $W_2 = \sum_{i=1}^{N_S} w_i^2$.