Complete Question
The complete question is shown on the first and second uploaded image
Answer:
The marginal density function of Y is [tex]f(y)= \frac{e^{\frac{-y}{10} }}{10}[/tex]
The distribution is exponential distribution
and the expected value is[tex]E[Y] = 10[/tex]
Step-by-step explanation:
From the question we are told that the function is
[tex]f(x,y) = \frac{e^{-\frac{y}{10} }}{10y} \ \ \ 0< y <x<2y[/tex]
Now the marginal density function of Y i.e f(y) is mathematically evaluated by obtaining the probability density function of y as follows
[tex]= \int\limits^{2y}_{y} { \frac{e^{\frac{-y}{10} }}{10y} } \, dx[/tex]
[tex]= \frac{e^{\frac{-y}{10} }}{10y} * (2y - y )[/tex]
[tex]= \frac{e^{\frac{-y}{10} }}{10}[/tex]
The distribution of the function above is exponential distribution with a rate parameter equals to
[tex]\lambda = \frac{1}{10}[/tex]
The mean DOT estimate E{Y} is mathematically evaluated as
[tex]E[Y] = \frac{1}{\lambda}[/tex]
substituting value
[tex]E[Y] = \frac{1}{\frac{1}{10} }[/tex]
[tex]E[Y] =10[/tex]
