This question is from "Mathematical Statistics with application by Wackerly and Mendelhall" $8th$ edition page $233$ ,question $5.5$ :
$f(y_1,y_2)=3y_1 , 0 \leq y_2 \leq y_1 \leq1$ and $0$ , elsewhere.
Find $F(1/2 ,1/3) =P(Y_1 \leq 1/2,Y_2 \leq 1/3)$.
I am beginner in statistics , so there are something i could not understand.I tried to solve this question ,but my answer and answer key is different.First of all , i want you to check my solution to see what i am missing.
$\mathbf{\text{MY ATTEMPT:}}$
$$\int_{0}^{1/2}\int_{0}^{1/3}3y_1dy_2dy_1 =\int_{0}^{1/2}\bigg(\int_{0}^{1/3}3y_1dy_2\bigg)dy_1$$
$$\bigg(\int_{0}^{1/3}3y_1dy_2\bigg) =3y_1\times(1/3) - 3y_1\times (0)=y_1$$
$$\int_{0}^{1/2}y_1dy_1 =\frac{y_1^2}{2} \rightarrow \frac{1}{8} - 0=0.125$$
However , the answer key says that answer is $0.1065$
What am i missing.
I want to add something such that when we solve continuous bivarite probablities , is $ 0 \leq y_2 \leq y_1 \leq1$ equal to $0 <y_2 < y_1 <1$ or $0 <y_2 \leq y_1 <1$ or $0 \leq y_2 < y_1 <1$ or $0 <y_2 < y_1 \leq1$ .I always confuse when i see $\leq$ and $<$ ,somebooks use $<$ , but somebooks use $\leq$.Can you also explain how to approach these signs in bivarite probability. Thanks in advance..
By the way , do not forget that i am a beginner , so please explain like explaining to an dummy..
The rule of $3y_1$ only holds when $0 \le y_2 \le y_1 \le 1$, it takes zero otherwise.
Let's describe the region, what are the values that $y_2$ can take? It takes value between $0$ and $\frac13$.
What are the values that $y_1$ can take in term of $y_2$, it takes value between $y_2$ and $\frac12$.
The limit of the integral should be
\begin{align} \int_0^\frac13 \int_{y_2}^\frac12 3y_1 \,\, dy_1 dy_2 &= \int_0^\frac13 \frac32y_1^2|_{y_2}^\frac12 \, dy_2 \\ &= \frac32\int_0^\frac13 \frac14 - y_2^2 \, dy_2\\ &= \frac32 \left(\frac1{12} - \frac1{81} \right)\\ &=\frac12 \left(\frac14 -\frac1{27}\right) \end{align}
For continuous variable, the random variable takes a particular value with probability $0$, hence the equality does not matter.