Let $X$ and $Y$ be continuous random variables with probability density functions $P_X(x)$ and $P_Y(y)$, respectively. Further, let $Y = X^2$ and $P_X(x) = \begin{cases} 1, & x\in (0,1] \\ 0, & \text{otherwise} \end{cases}$
Which one of the following options is correct?
$P_Y(y) = \begin{cases} \frac{1}{2\sqrt{y}}, & y\in (0,1] \\ 0, & otherwise \end{cases} $
Solution: Finding the PDF of Y = X^2
We are given the probability density function (PDF) of a continuous random variable X:
\( P_X(x) = \begin{cases} 1, & x \in (0,1] \\ 0, & \text{otherwise} \end{cases} \)
We are also given the relationship \( Y = X^2 \). We need to find the PDF of Y, denoted as \( P_Y(y) \).
1. Determine the Range of Y
Since \( X \) is defined on \( (0, 1] \), and \( Y = X^2 \), the range for \( Y \) is found by squaring the bounds of \( X \):
\( 0^2 < Y \le 1^2 \)
Therefore, \( Y \) is defined on the interval \( (0, 1] \).
2. Calculate the Cumulative Distribution Function (CDF) of Y
The CDF of \( Y \) is \( F_Y(y) = P(Y \le y) \).
Substitute \( Y = X^2 \):
\( F_Y(y) = P(X^2 \le y) \)
Since \( X > 0 \) and we are considering \( y > 0 \), this is equivalent to:
\( F_Y(y) = P(X \le \sqrt{y}) \)
This equals the CDF of \( X \) evaluated at \( \sqrt{y} \), i.e., \( F_X(\sqrt{y}) \).
First, find the CDF of \( X \), \( F_X(x) \). For \( x \in (0, 1] \):
\( F_X(x) = \int_{-\infty}^{x} P_X(t) dt = \int_{0}^{x} 1 dt = [t]_0^x = x \)
So, \( F_X(x) = x \) for \( x \in (0, 1] \).
Now, substitute \( \sqrt{y} \) into \( F_X(x) \). Since \( y \in (0, 1] \), \( \sqrt{y} \) is also in \( (0, 1] \).
\( F_Y(y) = F_X(\sqrt{y}) = \sqrt{y} \quad \text{for } y \in (0, 1] \)
3. Derive the PDF of Y from its CDF
The PDF \( P_Y(y) \) is the derivative of the CDF \( F_Y(y) \):
\( P_Y(y) = \frac{d}{dy} F_Y(y) \)
For \( y \in (0, 1] \):
\( P_Y(y) = \frac{d}{dy} (\sqrt{y}) = \frac{d}{dy} (y^{1/2}) = \frac{1}{2} y^{(1/2 - 1)} = \frac{1}{2} y^{-1/2} = \frac{1}{2\sqrt{y}} \)
Therefore, the PDF of \( Y \) is:
\( P_Y(y) = \begin{cases} \frac{1}{2\sqrt{y}}, & y \in (0,1] \\ 0, & \text{otherwise} \end{cases} \)
Important Questions from Random Variables
- Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained.
The expectation of $X$ is __________ (rounded off to two decimal places). - Let $X = aZ + b$, where $Z$ is a standard normal random variable, and $a, b$ are two unknown constants. It is given that
$E[X] = 1$, $E[(X – E[X])Z] = –2$, $E[(X – E[X])^2] = 4$,
where $E[X]$ denotes the expectation of random variable $X$. The values of $a, b$ are: Let $Y = Z^2$, $Z = \frac{X - \mu}{\sigma}$, where $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. The variance of $Y$ is
Consider a discrete random variable X whose probabilities are given below. The standard deviation of the random variable is ________ (round off to one decimal place).
$x_1$ 1 2 3 4 $P(X = x_i)$ 0.3 0.1 0.3 0.3 If $X$ is a continuous random variable with the probability density function
$f(x) = \begin{cases} \frac{K}{4}, & 0 \le x \le 1 \\ 0, & \text{otherwise} \end{cases}$
then the value of $K$ is __________. (Answer in integer)
