Let $(X,Y)$ be a random vector with probability density function. Find marginal density and probabilities

Question

Let $(X,Y)$ be a random vector with probability density function:

$\;f(x,y)= \left\{ \begin{array}{ll} \frac15, & for\;(x,y)\in [0,5]\times [0,1]\\ 0, & otherwise \\ \end{array} \right. $

$(a)$ find marginal density of $X$

$(b)$ $P(Y<\frac12)=$?

$(c)$ find if $X$ and $Y$ are independent

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$(a)$ $f_X(x)=\int_R f(x,y)\;dy = \int_0^1 \frac15dy = \frac15 \mathbf{1}_{[0,1]}(x)$

$(b)$ $P(Y< \frac12)=\int_0^{\frac12} \frac15dy=\frac15(\frac12)=\frac{1}{10}$

$(c)$ ?

Please help with $(c)$ and check the above. Will be grateful.