# Daily Archives: June 9, 2008

## The Bernoulli Shortcut

If $X$ has a Standard Bernoulli Distribution, then it can only have values 0 or 1 with probabilities $q$ and $1-q$.  Any random variables that can only have 2 values is a scaled and translated version of the standard bernoulli distribution.

Expected Value and Variance:

For a standard bernoulli distribution, $E[X] = q$ and $Var(X) = q(1-q)$.  If $Y$ is a random variable that can only have values $a$ and $b$ with probabilities $q$ and $(1-q)$ respectively, then

$\begin{array}{rl} Y &= (a-b)X +b \\ E[Y] &= (a-b)E[X] +b \\ Var(Y) &= (a-b)^2Var(X) \\ &= (a-b)^2q(1-q) \end{array}$

Filed under Probability

## Normal Approximation

If a random variable $Y$ is normal, you can map it to a standard normal distribution $X$ (useful for finding probabilities in the standard normal table) by the following relationship:

$Y = \mu_y + \sigma_yX$

Example 1:  $Y$ is normal.  $E[Y] = 100$ and $Var(Y) = 49$  Then

$\begin{array}{rl} P(Y \leq 111.515) &= P(X \leq \frac{111.515 - 100}{\sqrt{49}}) \\ &= P(X \leq 1.645) \\ &= 0.95 \end{array}$

Example 2:  $Y$ has the same distribution as example 1.  Then $P(Y \leq y) = 0.9$ implies

$P(X \leq \frac{y - 100}{\sqrt{49}}) = 0.9$

Which implies:

$\frac{y - 100}{\sqrt{49}} = 0.8159$

Hence $y = 105.7113$.

With regard to Central Limit Theorem:

By the Central Limit Theorem, the distribution of a sum of iid random variables converges to a normal distribution as the number of iid random variables increases.  This means that if the number of iid random variables is sufficiently large, we can get approximate probabilities by using a normal distribution approximation.