Probability of continuous random variables

In summary: You must find P(Y\le 2).In summary, the conversation involves discussing a piecewise function with different constants and finding probabilities related to it. One question is about finding P(Y=2) and the other is about finding the median of the distribution function. There is confusion about the correct approach to these questions, as well as uncertainty about the intended meaning of the given function.
  • #1
Gott_ist_tot
52
0

Homework Statement


A random variable has distribution function F(z) = P(y<= z) given by (this is a piecewise function)

f(z) =
0 if z < -1
1/2 if -1 <= z < 1
1/2 + 1/4(z-1 if 1 <= z < 2
1 if 2 <= z

What is P(Y = 2)?

Find all the numbers t with the property that both P(Y <= t) >= 1/2 and P(Y >= t) >= 1/2


Homework Equations





The Attempt at a Solution



For the P(Y=2) I integrated at the point 2 plus and minus epsilon and came up with 1/4z - 0 where z =2. Thus, 1/2. My concern is that this problem has a lot of constants Thus, I would expect P(Y=x) to equal 0 everywhere but in [1,2). Then I have no idea how to find the median of the distribution function. Sorry, if these are easy questions. The class is being taught without a book and I'm afraid I'm not used to that.
 
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  • #2
Gott_ist_tot said:
For the P(Y=2) I integrated at the point 2 plus and minus epsilon and came up with 1/4z - 0 where z =2. Thus, 1/2.
I don't understand what you're doing -- could you spell it out? The answer is certainly wrong.

My concern is that this problem has a lot of constants Thus, I would expect P(Y=x) to equal 0 everywhere but in [1,2).
Not just a lot of flat-lines in the graph, but it's mostly continuous! P(Y=a) can only be positive if the graph of the cumulative distribution function has a discontinuity. (right?)

Then I have no idea how to find the median of the distribution function.
Do you know how to find P(Y<=t)?
 
  • #3
well, no expert here, but to find P(Y=2) shouldn't you integrate f(z) over (-infinity,2]?
 
  • #4
F(z) = P(y<= z)

what exactly is "y" here?

the CDF is supposed to be F(z) = P[Z <= z]
 
  • #5
loop quantum gravity said:
well, no expert here, but to find P(Y=2) shouldn't you integrate f(z) over (-infinity,2]?
He said that f was the cumulative distribution, not the probability density.
 
  • #6
Gott_ist_tot said:

Homework Statement


A random variable has distribution function F(z) = P(y<= z) given by (this is a piecewise function)

f(z) =
0 if z < -1
1/2 if -1 <= z < 1
1/2 + 1/4(z-1 if 1 <= z < 2
1 if 2 <= z
This is impossible. The integral of a probability density function over its entire domain must be 1. You obviously can't have "f(z)= 1 if 2<= z". It also cannot be a cumulative probabililty distribution which was my next guess. I have no idea what is intended here.

loop quantum gravity said:
well, no expert here, but to find P(Y=2) shouldn't you integrate f(z) over (-infinity,2]?
No, that would give P(Y[itex]\le 2[/itex]). With a continuous probability density, the probobability that Y is any specific number is 0.
 

What is a continuous random variable?

A continuous random variable is a type of random variable that can take on any value within a certain range. This means that there are an infinite number of possible outcomes, and the variable can take on both whole numbers and fractions.

How is the probability of a continuous random variable calculated?

The probability of a continuous random variable is calculated by finding the area under the curve of the probability density function (PDF) between two points. This can be done using integration or by using a table of values.

What is the difference between a discrete and continuous random variable?

A discrete random variable can only take on a finite number of values, while a continuous random variable can take on any value within a certain range. Discrete random variables are usually associated with counting problems, while continuous random variables are used to model real-world phenomena.

What is the relationship between the probability density function and the cumulative distribution function?

The probability density function (PDF) describes the probability of a continuous random variable taking on a specific value. The cumulative distribution function (CDF) describes the probability of the variable being less than or equal to a certain value. The CDF can be calculated by integrating the PDF.

How is the mean and variance of a continuous random variable calculated?

The mean of a continuous random variable is calculated by finding the area under the curve of the variable multiplied by its corresponding values, and then dividing by the total area. The variance is calculated by taking the squared difference between each value and the mean, multiplying it by the corresponding area under the curve, and then dividing by the total area.

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