Difficult Probability Density Function Question

• Saracen Rue
In summary: This is because the question is asking for the value of ##a## for which the integral of the probability density function over the domain ##[b,c]## is equal to one, and this will only be true for one specific value of ##a##.To find this value, we can differentiate the function ##h(x,a)## with respect to ##a## and set it equal to zero to find the maximum or minimum value of the function. Then, we can use a numerical method to find the value of ##a## that satisfies the equation given in the question. Once we have this value, we can use it to find the coordinates of the points ##Q## and ##R##, as well as the mean, variance, and
Saracen Rue

Homework Statement

A function, ##f\left(x\right)=\left|a^{\frac{\sin \left(x\right)}{\ln \left(ax\right)}}-\frac{x}{a}\right|##, intersects with another function, ##g\left(x\right)=\left|\frac{sin(ln(\sqrt{x}-\sqrt{a}))}{x^2-a^2}\right|##, at point ##Q(b,f(b))## and point ##R(c,f(c))##. A probability density function, ##p(x)=g(x)-f(x)##, can be formed over the domain ##[b,c]##, where ##0<b<c<2##.

Q1. Determine, correct to 16 decimal places;
(a) The value of the real-valued constant, ##a##
(b) The coordinates of the points ##Q## and ##R##
Q2. Calculate the mean, variance and standard deviation correct to 4 decimal places

Q3. Find correct to 1 decimal place; the percentage probability of the continuous random variable, ##X##, being within ##a^2## variances either side of the mean (i.e. ##Pr(μ-a^2⋅Var(X)≤X≤μ+a^2⋅Var(X))##)

Homework Equations

For PDfs: ##∫_{b}^{c}f(x)dx=1##

The Attempt at a Solution

I know that to find points ##b## and ##c## I need to solve ##f(x)=g(x)## for ##x##, however when I try to do this on my calculator it gives me an error message. I know that I should be getting ##b## and ##c## in terms of ##a##, thus ##∫_{b}^{c}p(x)dx=1## should have ##a## as the only unknown and it should be able to be solved for. After getting ##a##, I'd substitute the value into ##b## and ##c## to get those values as well, and then I'd be able to find ##f(b)## and ##f(c)## as well (giving me points ##Q## and ##R##).

To find the mean, variance and standard deviation I'd just solve the following:
##E(X)=∫_{b}^{c}x⋅p(x)dx##, ##Var(X)=∫_{b}^{c}x^2⋅p(x)dx## and ##σ=\sqrt{Var(X)}##

And for ##Q3## I'd solve this: ##Pr(μ-a^2⋅Var(X)≤X≤μ+a^2⋅Var(X))=∫_{μ-a^2⋅Var(X)}^{μ+a^2⋅Var(X)}p(x)dx##

I am fairly confident with my method for the most part (although I'd still appreciate somebody telling me if there's any errors in it), the main issue I'm having is simply not being able to solve the equations on my calculator. I'd be very thankful for anybody who has an alternative way for me to solve this problem :)

If we can identify a nice search space then it shouldn't be too hard to numerically find a suitable value of ##a##.

To start that process, we can observe that, if we want ##f(x)-g(x)## to be defined and continuous on ##b,c##, we must have ##1\leq a<b##. Can you see why that must be the case?

Given that, we can differentiate ##h\triangleq f-g## and set it to zero to try to find extrema, of which there will have to be at least one between ##b## and ##c##.

Or - quicker and easier if it's allowed, just write a short bit of code in your favourite number-crunching language to search for a value of ##a\in[1,2]## for which ##h(x)## attains a maximum or minimum in the interval ##[a,2]## and then for such an ##a## see if the curve of ##h## crosses the ##x## axis on either side of that extremum.

By the way, the question is worded in a way that seems to imply that there is only one possible value of ##a##. In fact there is a range of possible values. So you only need to find a value of ##a## that will work.

andrewkirk said:
If we can identify a nice search space then it shouldn't be too hard to numerically find a suitable value of ##a##.

To start that process, we can observe that, if we want ##f(x)-g(x)## to be defined and continuous on ##b,c##, we must have ##1\leq a<b##. Can you see why that must be the case?

Given that, we can differentiate ##h\triangleq f-g## and set it to zero to try to find extrema, of which there will have to be at least one between ##b## and ##c##.

Or - quicker and easier if it's allowed, just write a short bit of code in your favourite number-crunching language to search for a value of ##a\in[1,2]## for which ##h(x)## attains a maximum or minimum in the interval ##[a,2]## and then for such an ##a## see if the curve of ##h## crosses the ##x## axis on either side of that extremum.

By the way, the question is worded in a way that seems to imply that there is only one possible value of ##a##. In fact there is a range of possible values. So you only need to find a value of ##a## that will work.

I think the question is implying that there will be only one value of ##a## for which the equation ##1 = \int_{b(a)}^{c(a)} h(x,a) \, dx## will work.

1. What is a probability density function (PDF)?

A probability density function is a mathematical function that describes the probability that a random variable falls within a certain range of values. It is used to model continuous random variables and is often represented graphically as a curve.

2. What makes a probability density function difficult?

A difficult probability density function is one that is complex and cannot be easily solved using traditional methods. This may be due to the complexity of the function itself, or the need to integrate over a large number of variables.

3. How do you approach solving a difficult probability density function?

The first step in solving a difficult probability density function is to understand the problem and any given constraints. Then, various mathematical techniques such as integration, differentiation, and substitution can be used to simplify the function and make it easier to solve.

4. What are some common techniques used to solve difficult probability density functions?

Some common techniques used to solve difficult probability density functions include the change of variables technique, the method of moments, and the maximum likelihood estimation method. These techniques involve manipulating the function in various ways to make it easier to solve.

5. How can I check if my solution to a difficult probability density function is correct?

One way to check the correctness of your solution is to compare it to known results or solutions for similar problems. You can also use numerical methods, such as Monte Carlo simulation, to approximate the solution and compare it to your analytical solution. Additionally, double-checking your calculations and ensuring that all assumptions and constraints are accounted for can help verify the accuracy of your solution.

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