A student studies for X number of hours during a randomly selected school day. The probability that X can…

Probability of studying at most two hours

The distribution must satisfy $\sum P(X = x) = 1$, where $P(X=0) = 0.2$, $P(X=1) = k$, $P(X=2) = 2k$, $P(X=3) = k(6-3) = 3k$, and $P(X=4) = k(6-4) = 2k$.

Summing yields $0.2 + k + 2k + 3k + 2k = 0.2 + 8k = 1$, so $8k = 0.8$ and $k = 0.1$.

The probability $P(X \le 2) = P(X=0) + P(X=1) + P(X=2) = 0.2 + 0.1 + 0.2 = 0.5$.

$\boxed{0.5}$