@ -410,7 +411,7 @@ To get probability from a to b (inclusive and exclusive doesn't matter in contin
+ Mean
+ Mean
\[ E(aX) = aE(X) \]
\[ E(aX) = aE(X) \]
\[ E(a) = a \]
\[ E(a) = a \]
\[ E(X + Y) = E(X) + E(Y) ]
\[ E(X + Y) = E(X) + E(Y) \]
+ Variance
+ Variance
If
If
@ -576,3 +577,159 @@ Given,
\[ X_2 \sim N(\mu_2, \sigma_2) \]
\[ X_2 \sim N(\mu_2, \sigma_2) \]
Then,
Then,
\[ a X_1 + b X_2 \sim N \left( a \mu_1 + b \mu_2, \sqrt{ a^2 \sigma_1^2 + b^2 \sigma_2^2} \right) \]
\[ a X_1 + b X_2 \sim N \left( a \mu_1 + b \mu_2, \sqrt{ a^2 \sigma_1^2 + b^2 \sigma_2^2} \right) \]
* Standard Normal Distribution
The normal distribution with Mean 0 and Variance 1 is called the standard normal distribution.
\[ Z \sim N(0,1) \]
To calculate area under a given normal distribution, we can use the standard normal distribution. For that we need to calculate corresponding values in standard distribution from our given distribution. For that we have formula
\[ For\ X \sim N(\mu, \sigma) \]
\[ z = \frac{x - \mu}{\sigma} \]
\[ x \rightarrow value\ in\ our\ normal\ distribution \]
\[ \mu \rightarrow mean\ of\ our\ distribution \]
\[ \sigma \rightarrow standard\ deviation\ of\ our\ distribution \]
\[ z \rightarrow corresponding\ value\ in\ standard\ normal\ distribution \]
Example,
Suppose for a normal distribution with X \sim N(\mu, \sigma) and we want to calculate probability P(a < X < b), then the ranges for same proability in the Z normal distribution will be,