Linear Transformation of a Single Random Variable
- In probability theory and statistics, a linear transformation of a random variable is a fundamental concept that allows us to manipulate and analyze data more effectively.
- For a random variable $X$, a linear transformation takes the form $aX + b$, where $a$ and $b$ are constants.
Expected Value of a Linear Transformation
The expected value of a linear transformation of a random variable $X$ is given by:
$$E(aX + b) = aE(X) + b$$
NoteThis formula is incredibly useful as it allows us to calculate the expected value of transformed data without having to recalculate the entire distribution.
ExampleSuppose we have three random variables:
\[\begin{aligned} &X_1: \text{ Number of cars sold }, \quad E(X_1) = 10 \\&X_2: \text{ Number of motorcycles sold }, \quad E(X_2) = 5 \\&X_3: \text{ Number of bicycles sold }, \quad E(X_3) = 20 \end{aligned} \]
If the profit for each is 1000,500 , and $\$ 200$ respectively, the expected total profit would be:
$$E\left(1000 X_1+500 X_2+200 X_3\right)$$ $$=1000 E\left(X_1\right)+500 E\left(X_2\right)+200 E\left(X_3\right)$$ $$=1000(10)+500(5)+200(20)=10000+2500+4000=16,500 \$ $$
Variance of a Linear Transformation
The variance of a linear transformation of a random variable $X$ is given by:
$$VAR(aX + b) = a^2 VAR(X)$$
HintNotice that the constant $b$ doesn't affect the variance, as it doesn't contribute to the spread of the data.
NoteThe standard deviation of a linear transformation is the absolute value of $a$ times the standard deviation of $X$: $SD(aX + b) = |a| * SD(X)$
ExampleContinuing with our height example, if $VAR(X) = 100$ cm², then:
$VAR(0.3937X + 2) = 0.3937^2 * 100 ≈ 15.5$ inches²
Expected Value of Linear Combinations of n Random Variables
- When dealing with multiple random variables, we often need to consider their linear combinations.
- The expected value of a linear combination of $n$ random variables is:
$$E(a_1X_1 + a_2X_2 + ... + a_nX_n) = a_1E(X_1) + a_2E(X_2) + ... + a_nE(X_n)$$
HintThis property is known as the linearity of expectation and is extremely useful in many applications.
ExampleSuppose we have three random variables:
- $X_1$: Number of cars sold (E(X_1) = 10)
- $X_2$: Number of motorcycles sold (E(X_2) = 5)
- $X_3$: Number of bicycles sold (E(X_3) = 20)
If the profit for each is $1000, $500, and $200 respectively, the expected total profit would be:
$E(1000X_1 + 500X_2 + 200X_3) = 1000E(X_1) + 500E(X_2) + 200E(X_3)$ $= 1000(10) + 500(5) + 200(20) = 10000 + 2500 + 4000 = $16,500$
