Point Estimation
\(\bullet\)What is point estimation?
Example:
\(\bullet\) Bevan, Kullberg, and Rice ( 1979) studied random fluctuations of current across a muscle cell membrane. The cell membrane contained a large number of channels, which opened and closed at random and were assumed to operate independently. The net current resulted from ions flowing through open channels.
\(\bullet\) They obtained \(49,152\) observations of the net current, \(x_{1}, \ldots, x_{49152}\)
\(\bullet\) It seems appropriate to model the net current data,\(X_{1}, \ldots, X_{49152}i.i.d. N\left(\mu, \sigma^{2}\right),\) where \(\mu\) and \(\sigma^{2}\) represent the mean and variance of net
Quiestion: how to use the observed data, \(x_{1}, \ldots, x_{49152}\) to gain knowledge about the values of \(\mu\) and \(\sigma^{2} ?\)
\(\underline{Def:}\)(procedure of fitting a particular distribution to data)
1.observed data:$\underline{x_1,x_2,\dots,x_n}$
2.statistical modeling: