For the discrete random variable $X$ with the given probability mass function (PMF), let's organize the data first:

$x$	1	2	3	4
$P(X=x)$	0.1	0.3	0.4	0.2

Key concepts related to a discrete random variable like this include:

1. Expected value (mean):
   The expected value $E(X)$ is calculated as: $E(X) = \sum_{x} x \cdot P(X = x)$ Let's compute this.

2. Variance:
   The variance $\text{Var}(X)$ measures the spread of the values around the mean and is calculated as: $\text{Var}(X) = E(X^2) - (E(X))^2$ where $E(X^2)$ is the expected value of $X^2$, given by: $E(X^2) = \sum_{x} x^2 \cdot P(X = x)$

3. Standard deviation:
   This is the square root of the variance: $\sigma_X = \sqrt{\text{Var}(X)}$

I'll now compute the expected value, variance, and standard deviation for $X$.Here are the computed values for the discrete random variable $X$:

1. Expected value $E(X)$: 2.7
2. Variance $\text{Var}(X)$: 0.81
3. Standard deviation $\sigma_X$: 0.9

Would you like further details on these calculations or have any questions?

Related questions:

1. How is the expected value useful in understanding random variables?
2. What does the variance tell us about the distribution of $X$?
3. How does standard deviation relate to variance in discrete distributions?
4. How can we calculate the skewness of a discrete random variable?
5. What is the interpretation of the PMF graph for a discrete random variable?

Tip:

The expected value provides the long-term average of a random variable if the experiment is repeated many times.