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As a measure of intelligence, mice are timed when going
Chapter 4, Problem 15E(choose chapter or problem)
As a measure of intelligence, mice are timed when going through a maze to reach a reward of food. The time (in seconds) required for any mouse is a random variable with a density function given by
\(f(y)=\left\{\begin{array}{ll}
\frac{b}{y^{2}}, & y \geq b \\
0, & \text { elsewhere }
\end{array}\right.\)
where b is the minimum possible time needed to traverse the maze.
a Show that \(f(y)\) has the properties of a density function.
b Find \(F(y)\)
c Find \(P(Y>b+c)\) for a positive constant c.
d If c and d are both positive constants such that \(d>c\), find \(P(Y>b+d \mid Y>b+c)\).
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QUESTION:
As a measure of intelligence, mice are timed when going through a maze to reach a reward of food. The time (in seconds) required for any mouse is a random variable with a density function given by
\(f(y)=\left\{\begin{array}{ll}
\frac{b}{y^{2}}, & y \geq b \\
0, & \text { elsewhere }
\end{array}\right.\)
where b is the minimum possible time needed to traverse the maze.
a Show that \(f(y)\) has the properties of a density function.
b Find \(F(y)\)
c Find \(P(Y>b+c)\) for a positive constant c.
d If c and d are both positive constants such that \(d>c\), find \(P(Y>b+d \mid Y>b+c)\).
Step 1 of 4
a).
Now we have to show that f(y) has the property of density function.
We have been asked to proof the property of distribution.
For \(b \geq 0, f(y) \geq 0\).
We are integrating f(y).
So the limits is \(-\infty \text { to }+\infty\).
Then,
\(\begin{aligned}
\int_{-\infty}^{+\infty} f(y) & =\int_{-\infty}^{+\infty} \frac{b}{y^{2}} \\
= & {[-b y]_{b}^{\infty} } \\
= & 1
\end{aligned}\)
Therefore f(y) has the property of density function is 1.
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