Random Variable and Distribution Function Relationship

In summary, we have proven that G(θ) has the same distribution function as F, and that F(X) has a uniform distribution on [0,1]. We have also addressed the question of what happens if F is not continuous, and shown that it still holds in that case. This may seem confusing at first, but remember to focus on the definitions and properties of distribution functions and random variables in order to understand these concepts better.
  • #1
shoeburg
24
0
I'm in a probability theory class and I feel like I'm missing something fundamental between random variables and their distribution functions. I was given the following questions:


1)Let θ be uniformly dist. on [0,1]. For each dist. function F, define G(y) = sup{x:F(x)≤y}. Prove G(θ) has the dist. function F.

My attempt: G(θ)=sup{x:F(x)≤θ}, which is saying G(θ) equals the biggest x that satisfies F(x)≤θ, which isn't too surprising since 0≤ F(x),θ ≤1. So for any value of θ, which is any value from [0,1], G(θ)=sup{x : Pr(ω : X(ω) ≤ x ) ≤ θ}, and this is where I get stuck. It's weird I feel like I totally believe the statement intuitively, but I can't figure out how to prove/explain it, so I think I'm missing out some details between dist. functions and random variables.

2) Let X have the continuous dist. function F. Prove F(X) has the uniform dist. on [0,1].

My attempt: I guess F(X) a composite function, taking sample space points to the reals, then to [0,1]? Or does F(X) = Pr(X≤X)? But that doesn't make sense to me. The problem also asked what if F is not continuous.

If anyone can help me out either in or outside the context of these problems, I'd really appreciate it. Maybe I'm getting too caught up thinking of what spaces these functions are mapping what into what, and measures and sigma-algebras and whatnot... Prob. Theory is kicking my butt haha
 
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  • #2
. Thanks in advance! For 1), let's start with the definition of G(θ): G(θ)=sup{x:F(x)≤θ}. This basically says that G(θ) is the largest x such that F(x)≤θ. Therefore, if we look at the probability that G(θ) is less than or equal to some value y, it will be equal to the probability that F(x)≤θ for some x less than or equal to y. In other words, it will be equal to the probability that F(x)≤y. Since F is a distribution function, this probability is equal to F(y), which means that G(θ) has the same distribution function as F. For 2), let's start by restating the problem: we want to prove that F(X) has a uniform distribution on [0,1]. We can do this by proving that F(X) has the same distribution function as a uniform random variable U on [0,1]. To do this, we need to show that for any y in [0,1], Pr(F(X)≤y) = Pr(U≤y). First, let's consider Pr(F(X)≤y). Since F is a distribution function, this is equal to Pr(X≤F-1(y)), where F-1 is the inverse of F. Now let's consider Pr(U≤y). This is equal to y, since U is uniform on [0,1]. Therefore, Pr(F(X)≤y) = Pr(X≤F-1(y)) = y = Pr(U≤y). This shows that F(X) has the same distribution function as U, and thus has a uniform distribution on [0,1].
 

FAQ: Random Variable and Distribution Function Relationship

What is a random variable?

A random variable is a numerical quantity that is determined by chance or uncertainty. It represents the possible outcomes of a random experiment or process.

What is the relationship between a random variable and a distribution function?

A distribution function is a mathematical function that describes the probability of different outcomes of a random variable. It shows how likely each outcome is to occur. The relationship between a random variable and a distribution function is that the distribution function is used to calculate the probability of a random variable taking on a specific value or falling within a certain range of values.

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

A discrete random variable can only take on a finite or countably infinite number of values, while a continuous random variable can take on any value within a given range. This means that a discrete random variable can be counted, while a continuous random variable can be measured.

How is the mean of a random variable related to its distribution function?

The mean of a random variable is the average value that the variable takes on. It is related to the distribution function by the expected value, which is calculated by multiplying each possible outcome of the random variable by its probability and summing them all together. The expected value is equal to the mean of the random variable.

What is the central limit theorem and how does it relate to random variables and distribution functions?

The central limit theorem states that the sum of a large number of independent and identically distributed random variables will follow a normal distribution, regardless of the distribution of the individual variables. This theorem is important because it allows us to make predictions and draw conclusions about a population based on a sample of data. It also highlights the relationship between random variables and their distribution functions, as the distribution of the sum of random variables is determined by their respective distribution functions.

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