Q.A random variable $X$ has the following probability distribution: $P(X=0)=0.1$, $P(X=1)=k$, $P(X=2)=2k$, $P(X=3)=0.3$. Find the value of $k$ and hence write the complete probability distribution.

A random variable is a real-valued function defined on the sample space of a random experiment, assigning a single number to every outcome. A discrete random variable takes finitely many distinct values $x_1,\dots,x_n$, and its probability distribution is the table pairing each value with its probability $p_i=P(X=x_i)$. A valid probability distribution must satisfy two conditions: every $p_i\ge0$, and the probabilities sum to exactly $1$ ($\sum p_i=1$). Constructing a distribution from a word problem means listing every possible value, computing each value's probability from the underlying sample space, and checking the sum-to-one condition as a safeguard.