Probability Density Functions

[Dollydaggerxo]

Statistics

Jill is expecting a vist from her uncle Tom. Tom would just come whenever he felt like it, but not when he had his art class. He will either visit her in the morning, or late afternoon, between the times of 9-12am or between 3-5pm. Say X is the number of hours after 7am, what would f(x) be and what would the sketch look like?

I have this qeustion on my homework. There are other similar, but they don't have two time intervals, so i don't know what to do.

I have draw the sketch of the probability density function as just two rectangles, between 2 and 5 and then between 8 and 10, with a height of 1/5. but I am not sure if this is correct, should it be curved? i didnt think so, as the probabilities are equal for all.

i'm stuck on the f(x) bit too.

 

[Chewy0087]

You're right about the sketch, the proper name for it is the uniform distribution.

When it asks for f(x), what it's asking for is the probability distribution function, where you define the graph for all values of x, for example;

f(x) = 0 (for x < 0) , 1 (for 0 < x < 5), 0 (for x > 5)

or something similar to that.

 

[Architeuthis Dux]

Probability density functions (PDFs) are mathematical representations of the likelihood that a random variable falls within a particular range of values. In this case, the random variable is the time of Tom's visit, and the range of values is between 9-12am and 3-5pm. The PDF for this situation would be a step function, with a value of 1/5 between 2 and 5, and again between 8 and 10, and a value of 0 everywhere else. This represents the equal probability of Tom visiting during either time interval.

In terms of the sketch, you are correct in drawing two rectangles with a height of 1/5. However, it would be more accurate to represent the rectangles as having a width of 3 (for the morning interval) and 2 (for the afternoon interval), as this corresponds to the time intervals of 9-12am and 3-5pm respectively.

As for the f(x) notation, this is simply the equation for the PDF. In this case, it would be f(x) = 1/5 for x between 2 and 5, and f(x) = 1/5 for x between 8 and 10. For all other values of x, f(x) = 0. This notation is used to mathematically describe the probability distribution and make calculations easier.

I hope this helps clarify the concept of probability density functions for you. Remember, PDFs are just one way to represent the likelihood of different outcomes in a statistical scenario, and there may be other ways to approach this problem as well. it's important to understand different statistical tools and choose the most appropriate one for a given situation.