# Probability/Random variables question

• ashah99
In summary, The conversation is about a probability question involving a uniform distribution and finding the area of a quarter of a circle with radius 1. The approach is to calculate the ratio of areas or use polar coordinates. The answer is correct and can be obtained using either method.
ashah99
Homework Statement
Finding the probability of randomly choosing a point within the unit square constrained within the quarter circle
Relevant Equations
P(( X, Y ) ∈ A) = ∫∫ fXY ( x, y )dxdy
Hello all, I am wondering if my approach is coreect for the following probability question? I believe the joint PDF would be 1 given that the point is chosen from the unit square. To me, this question can be reduced down to finding the area of 1/4 of a circle with radius 1. Any help is appreciated!

It's clear from your calculations that you are indeed simply calculating the area. Which is what you would expect for a uniform distribution.

PeroK said:
It's clear from your calculations that you are indeed simply calculating the area. Which is what you would expect for a uniform distribution.
Ok, makes sense. Would you agree that my answer is correct? Just want to make sure I understand.

ashah99 said:
Ok, makes sense. Would you agree that my answer is correct? Just want to make sure I understand.
Yes, it's just a ratio of areas, as you've calculated.

ashah99
Or use polar coordinates: $$\int_0^1 \int_0^{\sqrt{1-x^2}} 1\,dy\,dx = \int_0^{\pi/2}\int_0^1 r\,dr\,d\theta.$$

ashah99

## 1. What is probability?

Probability is a measure of the likelihood that a certain event will occur. It is represented as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

## 2. What is a random variable?

A random variable is a numerical outcome of a random experiment. It can take on different values with certain probabilities, and is used to model uncertain events in probability theory and statistics.

## 3. What is the difference between discrete and continuous random variables?

Discrete random variables can only take on a finite or countably infinite number of values, while continuous random variables can take on any value within a certain range. For example, the number of children in a family is a discrete random variable, while the height of a person is a continuous random variable.

## 4. How do you calculate the expected value of a random variable?

The expected value of a random variable is the sum of each possible value multiplied by its corresponding probability. This can be written as E(X) = ∑ x * P(X=x), where X is the random variable and x is a possible value it can take on.

## 5. What is the central limit theorem?

The central limit theorem states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the underlying distribution of the individual variables. This is a fundamental concept in statistics and allows for the use of normal distribution in many real-world scenarios.

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