GATE Probability and Statistics formula sheet

Selecting and arranging r of n distinct objects. P(n,n) = n! is the number of orderings of all n, and P(n,0) = 1 because 0! = 1.

Selecting r of n distinct objects with no arrangement. Related to permutations by P(n,r) = r! times the binomial coefficient. The binomial coefficient is taken as 0 when r > n.

Defined when $P(B)>0$. For fixed $B$, $P(\cdot \mid B)$ is itself a probability measure, so $P(A^c\mid B)=1-P(A\mid B)$.

The first form needs $P(B)>0$, the second needs $P(A)>0$. No independence is needed.

Raw data. For grouped data with frequencies $f_i$ and class midpoints $x_i$, use $\bar{x}=\sum f_i{x}_i/\sum f_i$. This treats every value as sitting at its class midpoint, so it is an approximation unless the values really are the midpoints.

Here $x_{(k)}$ is the $k$-th smallest value; sort first.

Use the second condition to find an unknown constant in a given PMF.

A density can exceed 1 at a point; only its integral is a probability.

Applies when each of the n consecutive integers is equally likely; zero elsewhere.

Here n = b - a + 1 and the values are consecutive integers. The variance depends only on n, not on where the block starts.

Use the total area equal to 1 to find an unknown constant in a given density.

F is non-decreasing and, for a continuous X, continuous. Its limits are 0 as x goes to -infinity and 1 as x goes to +infinity. The PDF is the derivative of the CDF wherever the derivative exists.

Needs independent observations with a common mean $\mu$ and finite variance ${\sigma }^2$; no normality is needed. SE (standard error) is the standard deviation of $\bar{X}$. When $\sigma$ is unknown it is estimated by $s/\sqrt{n}$, where $s$ is the sample standard deviation (divisor $n-1$, below).

Dividing by $n-1$ makes $S^2$ an unbiased estimator of ${\sigma }^2$. Dividing by $n$ instead gives an estimator whose expectation is $\frac{n-1}{n}{\sigma }^2$. Throughout these notes $s^2$ is the $n-1$ version.