Find the values of 'a' and 'b' for the following PDF

[Saracen Rue]

Homework Statement

A probability density function is defined by $f\left(x\right)=\left|a\right|\left(x+1\right)-\left(ax-1\right)^2$ where $x∈[0,b]$. Determine the values of the constants $a$ and $b$, given that the graph passes through the point $(b,0)$.

Homework Equations

$∫_a^bf(x)dx=1$ where $f(x)$ is a probability density function

The Attempt at a Solution

Okay, so I know I'm attempting to solve $∫_0^bf(x)dx=1$ for $b$, but I'm rather unsure of how to do that in this instance due to $a$ being a second unknown variable.

I've attempted finding the x-intercept, $b$ of $f(x)$ but it is dependent on the variable $a$. I then tried to find $a$ in terms of x and use that to somehow find $b$ but I soon realized that wasn't going anywhere. Normally for these types of questions, $a$ would just be a dilation factor, meaning it doesn't effect the values of the x-intercepts and can just be taken out as a common factor before integrating. I'm not sure why I'm supposed to do when it effects the location of the x-intercepts.

 

[Mark44]

Homework Statement

A probability density function is defined by $f\left(x\right)=\left|a\right|\left(x+1\right)-\left(ax-1\right)^2$ where $x∈[0,b]$. Determine the values of the constants $a$ and $b$, given that the graph passes through the point $(b,0)$.

Homework Equations

$∫_a^bf(x)dx=1$ where $f(x)$ is a probability density function

The Attempt at a Solution

Okay, so I know I'm attempting to solve $∫_0^bf(x)dx=1$ for $b$, but I'm rather unsure of how to do that in this instance due to $a$ being a second unknown variable.

I've attempted finding the x-intercept, $b$ of $f(x)$ but it is dependent on the variable $a$. I then tried to find $a$ in terms of x and use that to somehow find $b$ but I soon realized that wasn't going anywhere. Normally for these types of questions, $a$ would just be a dilation factor, meaning it doesn't effect the values of the x-intercepts and can just be taken out as a common factor before integrating. I'm not sure why I'm supposed to do when it effects the location of the x-intercepts.

With the integral you show in your attempt and the given information that f(b) = 0, you have two equations in the unknowns a and b. That should be enough information for you to solve for a and b.

 

[Saracen Rue]

With the integral you show in your attempt and the given information that f(b) = 0, you have two equations in the unknowns a and b. That should be enough information for you to solve for a and b.

Oh thank you I think I understand now. Because $b$ is the x-intercept, $f(x)=f(b)=0$, leaving us with just $a$ and $b$ as unknowns. As $∫_0^bf(x)dx=1$ also simplifies down to only containing $a$ and $b$. Thus, I should be able to solve $f(b)=0$ and $∫_0^bf(x)dx=1$ simultaneously to find both $a$ and $b$. Does this sound right?

 

[Mark44]

Oh thank you I think I understand now. Because $b$ is the x-intercept, $f(x)=f(b)=0$, leaving us with just $a$ and $b$ as unknowns. As $∫_0^bf(x)dx=1$ also simplifies down to only containing $a$ and $b$. Thus, I should be able to solve $f(b)=0$ and $∫_0^bf(x)dx=1$ simultaneously to find both $a$ and $b$. Does this sound right?

That's pretty much what I said.