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
Variance Calculation for Y = Z^2
We are given a normal random variable $X$ with mean $\mu$ and variance $\sigma^2$, denoted as $X \sim N(\mu, \sigma^2)$.
A standardized variable $Z$ is defined as $Z = \frac{X - \mu}{\sigma}$. This transformation makes $Z$ a standard normal random variable, meaning $Z \sim N(0, 1)$.
We need to find the variance of $Y = Z^2$.
Properties of Standard Normal Variable Z
- For $Z \sim N(0, 1)$, the expected value is $E[Z] = 0$.
- The variance is $Var(Z) = E[Z^2] - (E[Z])^2 = 1$.
- Since $E[Z] = 0$, we have $E[Z^2] - 0^2 = 1$, which means $E[Z^2] = 1$.
Relationship between Y and Chi-Squared Distribution
The variable $Y$ is defined as $Y = Z^2$. Since $Z \sim N(0, 1)$, the square of $Z$, which is $Z^2$, follows a Chi-squared distribution with 1 degree of freedom. This is denoted as $Y \sim \chi^2(1)$.
Calculating Variance of Y
The variance of a Chi-squared distribution with $k$ degrees of freedom ($\chi^2(k)$) is given by the formula $Var = 2k$.
In our case, $Y \sim \chi^2(1)$, so $k=1$. Applying the formula:
$ Var(Y) = 2k = 2 \times 1 = 2 $
Therefore, the variance of $Y$ is 2.
Important Questions from Random Variables
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?- 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: 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)
