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The 'mean' or 'expected value' of a random variable is a measure of its central tendency. Essentially, it tells us the average value we can expect if we repeat the experiment many times.
For a random variable X, the mean is denoted as \(\[\begin{equation} \mu = E[X] \end{equation}\]\).
When you add a constant K to the random variable X, it shifts the entire distribution by K, but the spread (or variability) around that mean remains the same. So, in mathematical terms:
\(\[\begin{equation} E[X + K] = E[X] + K = \mu + K \end{equation}\]\). This tells us that the new mean is \(\[\begin{equation} \mu + K \end{equation}\]\).
When you multiply the random variable X by a constant K, the mean scales by that constant. Here's how it's shown mathematically:
\(\[\begin{equation} E[KX] = K \cdot E[X] = K \mu \end{equation}\]\). This means the new mean is \(\[\begin{equation} K \mu \end{equation}\]\).

The variance of a random variable measures the spread (or dispersion) around its mean. It shows how much the values of the random variable deviate from the mean.
The variance of X is denoted as \(\[\begin{equation} \sigma^2 = Var(X) \end{equation}\]\).
Adding a constant K to the random variable X does not affect its spread. Hence, we have:
\(\[\begin{equation} Var(X + K) = Var(X) = \sigma^2 \end{equation}\]\).
However, multiplying the random variable X by a constant K affects the variance significantly. When X is multiplied by K, the variance scales by the square of K:
\(\[\begin{equation} Var(KX) = K^2 \sigma^2 \end{equation}\]\). This shows that the new variance becomes \(\[\begin{equation} K^2 \sigma^2 \end{equation}\]\).

The standard deviation is the square root of the variance and indicates the average distance of each data point from the mean. It is denoted as \(\[\begin{equation} \sigma = \sqrt{Var(X)} \end{equation}\]\).
Just like variance is unaffected by adding a constant to X, standard deviation is also unaffected:
\(\[\begin{equation} \sigma_{Y} = \sqrt{Var(Y)} = \sqrt{Var(X)} = \sigma \end{equation}\]\).
When X is multiplied by a constant K, the new standard deviation is scaled by K:
\(\[\begin{equation} \sigma_{Z} = K \sigma \end{equation}\]\).
This means if you multiply X by K, the spread (or variability) also multiplies by K.

Constant multiplication affects both the mean and the spread of the random variable.
When you multiply random variable X by a constant K:

• The new mean becomes \(\[\begin{equation} K \mu \end{equation}\]\).
• The new variance is \(\[\begin{equation} K^2 \sigma^2 \end{equation}\]\).
• The new standard deviation is \(\[\begin{equation} K \sigma \end{equation}\]\).

These transformations show a proportional increase or decrease, depending on the value of K, in both the central tendency and the spread. This scaling maintains the relative distribution's shape, but stretches or compresses it depending on K.

Constant addition only affects the location of the random variable distribution, moving it left or right but not altering its spread.
When you add a constant K to the random variable X:

• The new mean becomes \(\[\begin{equation} \mu + K \end{equation}\]\).
• The variance and standard deviation remain unchanged: \(\[\begin{equation} \sigma^2 \end{equation}\]\) and \(\[\begin{equation} \sigma \end{equation}\]\), respectively.
• This transformation only shifts the entire probability distribution by K.

Therefore, adding a constant changes the position of the distribution (location shift) but not its shape.