A discrete random variable X has the probability distribution given below:

Find the value of k.

Hint: In the probability distribution the sum of all probabilities is equal to 1. The four values are given in P(X). So, we simply do the sum of all P(X) values and then put it equal to 1. Using this we find the value of k.

Complete step-by-step answer:

When the value of X is 0.5, then the probability P(X) is k.
\[P(0.5)=k\].

When the value of X is 1, then the probability of this 2k.
$P(1)=2k$.

When the value of X is 1.5, then the probability of this 3k.
$P(1.5)=3k$.

When the value of X is 2, then the probability of this k.
$P(2)=k$.

Now, we have to find the value of k.

“The sum of the probabilities of all outcomes must be equal to 1”.
$\therefore $ If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. The probability that an event does not occur is 1 minus the probability that the event does occur.
The probability distribution sum of all probabilities is equal to 1.

Now,
Add all the values of P(X) and put it equal to 1.
$\begin{align}
  & \Rightarrow k+2k+3k+k=1 \\
 & \Rightarrow 7k=1 \\
 & \therefore k=\dfrac{1}{7} \\
\end{align}$

Therefore, the value of k in the probability distribution is $\dfrac{1}{7}$.

Note: The key step for solving this problem is the basic property of the sum of probability distribution for discrete random variable X. According to this property, the sum total of probabilities of all the events is one. This will provide us the required equation for solving and evaluating the value of k.