# The Probability Density of X^2?

• Kior
In summary, the conversation discusses finding the probability density of a uniformly distributed random variable using different methods. One method results in a different answer than the other. There is some confusion about the steps and equations used in the different methods. Ultimately, the correct answer is determined to be a density function of 0.5/√y.
Kior
Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out?

Question:
Let X be uniformly distributed random variable in the internal [ 0, 1]. Find the probability density of X^2?

My trial:
FY(y) = P(Y≤y) = P(X^2≤y) = P(X≤√y) = FX(√y) = ∫ (from 0 to √y) t dt = 0.5 y ⟹ 0.5 = fY(y).
This is actually inspired by http://math.stackexchange.com/questions/...

Solution on the textbook:
y = x^2
dy = 2x dx
h(y)dy = 1 dx
h(y) 2x dx = dx
h(y) = 0.5/x = 0.5/√y

Why t dt, not just dt? And your long line of equalities is bracketed with FY(y)=...=fY(y), which is clearly not true. Have you missed a step in typing it out?

haruspex said:
Why t dt, not just dt? And your long line of equalities is bracketed with FY(y)=...=fY(y), which is clearly not true. Have you missed a step in typing it out?

Thanks get it

Density function should be 1

haruspex said:
... And your long line of equalities is bracketed with FY(y)=...=fY(y), which is clearly not true. Have you missed a step in typing it out?
I suppose that we will never know ,

haruspex said:
Why t dt, not just dt? And your long line of equalities is bracketed with FY(y)=...=fY(y), which is clearly not true. Have you missed a step in typing it out?
They're not all equal signs; there's an arrow in there as well. The OP has ##F_Y(y) = y/2\ \Rightarrow\ f_Y(y) = 1/2##, which is, in fact, okay.

vela said:
They're not all equal signs; there's an arrow in there as well. The OP has ##F_Y(y) = y/2\ \Rightarrow\ f_Y(y) = 1/2##, which is, in fact, okay.
Ok. The ASCII character used for the arrow in the OP doesn't come out right on my iPad.

## 1. What is the probability density of X^2?

The probability density of X^2 is the likelihood of obtaining a particular value of X^2 in a continuous probability distribution. It is represented by the function f(x) and is used to calculate the probability of obtaining a value between two points on the distribution curve.

## 2. How is the probability density of X^2 calculated?

The probability density of X^2 is calculated by taking the derivative of the cumulative distribution function (CDF) of X^2. This results in the function f(x) which represents the probability density at any given point on the distribution curve.

## 3. What is the relationship between the probability density of X^2 and the standard normal distribution?

The probability density of X^2 is related to the standard normal distribution through the Chi-square distribution. This distribution is obtained by squaring and adding multiple independent standard normal variables, and its probability density is used to calculate the probability of obtaining a particular value of X^2.

## 4. How is the probability density of X^2 used in statistical analysis?

The probability density of X^2 is used in statistical analysis to calculate the likelihood of obtaining a particular value of X^2 in a given data set. It is commonly used in hypothesis testing, where the observed X^2 value is compared to the expected X^2 value to determine the significance of the results.

## 5. Can the probability density of X^2 be negative?

No, the probability density of X^2 cannot be negative. It represents the likelihood of obtaining a particular value of X^2, and since probability cannot be negative, the probability density must also be non-negative.

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