Probability in cdf

Question

A random variable Y has cdf:

$$F(y) = \begin{cases} 0 & y < 0 \\ \ln(y) & 1 \le y \le e \\ 1 & e < y \end{cases}$$

Find:

1. $P(Y < 2)$
2. $P(2 < Y < 2.5)$
3. $P(2 < Y \le 2.5)$
4. $f(y)$

For 4, I used differentiation to obtain $\dfrac{1}{y}$ for $1 \le y \le e$

Answer 1

1-

$P(Y<2) = P(Y<=2) - P(Y=2)$

$P(Y<2) = P(Y<=2) - 0$

$P(Y<2) = F(2) = ln(2)$

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2-

$P(2<Y<2.5) = P(Y < 2.5) - P(Y<2)$

$P(2<Y<2.5) = ln(2.5) - ln(2)$

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3-

$P(2<Y≤2.5) = P(Y < 2.5) - P(Y<2)$

$P(2<Y<2.5) = ln(2.5) - ln(2)$

4-

$\frac{1}{Y} , 1≤y≤e$