Question

If a random variable \(X\) has probability distribution \[ P(X=x)=kx, \] \[ x=1,2,3,4, \] then the value of \(k\) is:

• A. \(\frac15\)
• B. \(\frac1{10}\)
• C. \(\frac1{20}\)
• D. \(\frac1{15}\)

Hint:

Whenever an unknown constant appears in a probability distribution, use the fact that total probability equals \(1\).

Answer 1

The Correct Option is B

Concept: The sum of all probabilities in a probability distribution must be equal to \(1\). \[ \sum P(X=x)=1 \]

Step 1: Write all probabilities \[ P(1)=k \] \[ P(2)=2k \] \[ P(3)=3k \] \[ P(4)=4k \]

Step 2: Use total probability equals one \[ k+2k+3k+4k=1 \] \[ 10k=1 \]

Step 3: Solve for \(k\) \[ k=\frac1{10} \] Final Answer: \[ \boxed{\frac1{10}} \]