The probability distribution of a random variable \( X \) is given as \( P(X=x)=\begin{cases} kx^{3} & \text{for } x=1,2,3\\ 3kx & \text{for } x=4,5\\ 0 & \text{otherwise}\end{cases} \), where \( k \) is a constant.
The value of \( P(X\ge 4) \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{3}{7} \)
The probabilities must add up to 1:
\( \displaystyle k(1^{3}+2^{3}+3^{3})+3k(4+5)=1 \)
\( \displaystyle 36k+27k=1 \)
\( \displaystyle 63k=1 \)
\( \displaystyle k=\frac{1}{63} \)
Then
\( \displaystyle P(X\ge 4)=P(X=4)+P(X=5) \)
\( \displaystyle =3k(4)+3k(5) \)
\( \displaystyle =27k=\frac{27}{63} \)
\( \displaystyle =\frac{3}{7} \)