Let X denote the number of hours you study on a Sunday. Also it is known that $P(X = x) =\left\{\begin{matrix}0.1&, if\, x = 0\\kx&, if\, x = 1\, or\, 2\\k (5-x)&, if\, x = 3\, or\, 4\\0&,otherwise\end{matrix}\right.$ where k is a constant. What is the probability that you study atleast two hours? Exactly two hours? Atmost two hours?

$P(X≥2)=0.75, P(X=2)=0.30, P(X≤2)=0.55$

The correct answer is Option (1) → $P(X≥2)=0.75, P(X=2)=0.30, P(X≤2)=0.55$

From given information, we find that the probability distribution of X is

We know that $Σp_i = 1$

$⇒ 0.1+ k + 2k + 2k + k = 1⇒6k = 1-0.1 = 0.9$

$⇒k=0.15$

P(you study atleast two hours) = $P(X ≥2)$

$= P(2) + P(3) + P(4) = 2k + 2k + k$

$= 5k = 5 × 0.15 = 0.75$

P(you study exactly two hours) = $P(2) = 2k = 2 × 0.15 = 0.3$

P(you study atmost two hours) = $P(X ≤2) = P(0) + P(1) + P(2)$

$= 0.1+ k + 2k = 0.1 + 3k$

$= 0.1 + 3 × 0.15 = 0.55$