Answer to Question #212752 in Statistics and Probability for Tshimo

Question

1.  Consider the following joint probability density function (pdf) of the random variables X and Y:   f(x,y)  =  0 ≤ x ≤ 6,    0 ≤ y ≤ 8     =  0  elsewhere                                                                           

Calculate the constant c if P(2 < X < c, 3 < Y < 5) = 0,06                             

Answer 1

$$f(x, y)=\begin{cases} 1/48 & 0 ≤ x ≤ 6, 0 ≤ y ≤ 8 \\ 0 &elsewhere \end{cases}$$

Check

$$\displaystyle\int_{-\infin}^{{\infin}}\displaystyle\int_{-\infin}^{\infin}f(x, y)dydx=\displaystyle\int_{0}^{{6}}\displaystyle\int_{0}^{8}\dfrac{1}{48}dydx$$

$$=\dfrac{1}{48}\displaystyle\int_{0}^{{6}}[y]\begin{matrix} 8\\ 0 \end{matrix}dx=\dfrac{1}{6}[x]\begin{matrix} 6\\ 0 \end{matrix}=1$$

$$P(2<X<c, 3<Y<5)=\displaystyle\int_{3}^{5}\displaystyle\int_{2}^{c}\dfrac{1}{48}dxdy$$

$$=\dfrac{1}{48}\displaystyle\int_{3}^{5}[x]\begin{matrix} c \\ 2 \end{matrix}dy=\dfrac{c-2}{48}[y]\begin{matrix} 5 \\ 3 \end{matrix}=\dfrac{c-2}{24}=0.06$$

$$c=3.44$$