[Math] Find the constant K in probability distribution

Question

Find the constant K?
I know how to find K, if there is only one "k" in the question and the rest are given number. I know that all the x value are add upto 1.

But in this question, I don't know how to start.

One thing I know is $\Sigma(x)=1(k)+2(2k)+3(3k)+4(4k)+5(5k)$

Does this help to find the constant k?

Please help me "Find the constant k"
Appreciate your help.

Answer 1

You are confused. It's pretty fine. Please note that $X=i$ means the event that X equals i. So, let me interpret the table for you.

There's a probability of $k$ that event $X=1$ shall happen.
There's a probability of $2k$ that event $X=2$ shall happen.
There's a probability of $3k$ that event $X=3$ shall happen.
There's a probability of $4k$ that event $X=4$ shall happen.
There's a probability of $5k$ that event $X=5$ shall happen.
No other events are possible.

Important: By Unitarity of probability, the probability of $Pr({\text{$X$ = 1,2,3,4 or 5}})=1$.

$X=1, X=2, X=3, X=4, X=5$ are five mutually exclusive events in the sample space, and there are no other events. So, $$Pr(X=1,2,3,4,5)=\sum_{i=1}^5Pr(X=i)=k+2k+3k+4k+5k = 15k$$

So, what's $k$?