Statistics - Class 11

Let $f(x) = |2x^3 - 9x^2 + 12x - 3|$. If $M$ and $m$ are the global maximum and global minimum values of $f(x)$ on the interval $[0, 3]$ respectively, then the value of $M + m$ is

Consider the family of ellipses $E_t: \frac{x^2}{t} + \frac{y^2}{10-t} = 1$, where $t \in [2, 8]$. For each $t$, let $D_t$ be the diameter of $E_t$ that bisects all chords of slope $1$. Let $P_t$ be the point on $D_t$ whose abscissa is $\frac{t}{10}$. As $t$ varies, $P_t$ traces a path on a straight line $\Gamma$. For each $t$, let $C_t$ be the chord of $E_t$ that lies on the line $\Gamma$. Let $L(t)$ be the length of the chord $C_t$. If $R$ is the range of $L(t)$ and $C_R$ is the coefficient of range of $L(t)$ for $t \in [2, 8]$, find the value of $R \cdot C_R$.

Let $f(x)$ be the sum of the first 20 terms of the series $1 + 4(x-2) + 7(x-2)^2 + 10(x-2)^3 + \dots$ If $R$ is the range of $f(x)$ for $x \in [2, 3]$, find the coefficient of range of $R$.

Let $A(\theta) = \begin{pmatrix} \sin\theta & 1 & \cos\theta \\ \cos\theta & 2 & \sin\theta \\ -\sin\theta & 1 & \cos\theta \end{pmatrix}$. For all $\theta \in [0, 2\pi]$ where $A(\theta)$ is invertible, find the range of the function $f(\theta) = \det(A^{-1}(\theta))$.

Let a line $L$ be parallel to the vector $\vec{v} = (\cos \alpha - \sin \alpha)\hat{i} + (\cos \alpha + \sin \alpha)\hat{j} + \sqrt{6}\hat{k}$, where $\alpha \in [0, 2\pi)$. Let $\theta$ be the angle between the line $L$ and the plane $P: x + y + \sqrt{6}z = 2024$. If $R$ and $C_R$ denote the range and the coefficient of range of $\theta$ respectively, find the ordered pair $(R, C_R)$.

Let a discrete random variable $X$ take values from the set $S = \{0, 1, \frac{3}{2}, 2, 3\}$. The probability mass function of $X$ is given by $P(X=x) = \frac{2x+2}{25}$ for all $x \in S$. A new random variable is defined as $Y = \text{sgn}(X^2 - 3X + 2)$, where $\text{sgn}(\cdot)$ denotes the signum function. If $\alpha$ is the mean deviation of $Y$ about its mean, and $\beta$ is the mean deviation of $Y$ about its median, find the value of $100(\alpha - \beta)$.

Let $A$ be a $5 \times 5$ real matrix of rank $3$ such that $\text{Tr}(A) = 20$ and $A^4 - 20A^3 + 131A^2 - 280A = O$ Let $S$ be the data set consisting of all the eigenvalues of $A$ (including multiplicities). If $M_1$ and $M_2$ are the mean deviations of $S$ about its mean and median respectively, then the value of $45(M_1 - M_2)$ is:

Let $A = \begin{pmatrix} 2 & -1 & -1 & -1 \\ -1 & 2 & -1 & -1 \\ -1 & -1 & 2 & -1 \\ -1 & -1 & -1 & 2 \end{pmatrix}$ be a $4 \times 4$ matrix. Define a function $f: \mathbb{R} \to \mathbb{R}$ as $f(x) = \text{tr}(\text{adj}(xI - A))$, where $I$ is the $4 \times 4$ identity matrix. A statistics dataset $S$ is formed by evaluating $f(x)$ at the first five natural numbers, i.e., $S = \{f(1), f(2), f(3), f(4), f(5)\}$. If $M_1$ and $M_2$ denote the mean deviation about the mean and the mean deviation about the median of the dataset $S$ respectively, then the value of $5(M_1 - M_2)$ is:

Let $z_1, z_2, z_3, z_4$ be the roots of the polynomial equation $z^4 - \alpha z^3 + \beta z^2 - \gamma z + 1 = 0$, where $\alpha, \beta, \gamma \in \mathbb{C}$. Given that $|\alpha| = 4$, the arithmetic mean of the moduli of the roots is $1$, and $\text{Im}(\alpha) > 0$. The value of $|\beta + \gamma|^2$ is:

A subset $A$ of 4 distinct numbers is chosen uniformly at random from all subsets of $S = \{1, 2, 3, \dots, 15\}$ that have the property that no two elements in the subset sum to 15. Let $M$ be the expected value of the arithmetic mean of the elements in $A$. Find the value of $M$.

Let $r(x) = \frac{x^2 - 5x + 4}{x^2 + x - 2}$. Let $S(x)$ denote the sum of the infinite series $1 + 3r(x) + 5r(x)^2 + 7r(x)^3 + \dots$ If $S(x)$ is equal to the arithmetic mean of the functions $f(x) = \frac{4r(x)^2 + 18r(x)}{(1-r(x))^2}$ and $g(x) = \frac{2r(x) - 8}{(1-r(x))^2}$, then the sum of all valid values of $x$ is:

Let $a_1, a_2, a_3$ be the roots of the equation $x^3 - 3x^2 + 4x - 5 = 0$. Let $B$ be a $3 \times 3$ matrix defined as: $B = \begin{pmatrix} \frac{a_2+a_1}{2} & 0 & \frac{a_2+a_3}{2} \\ 0 & \frac{a_1+a_2}{2} & \frac{a_1+a_3}{2} \\ \frac{a_3+a_1}{2} & \frac{a_3+a_2}{2} & 0 \end{pmatrix}$ Find the value of $|B \cdot \text{adj}(-2B)|$.

Let $X$ and $Y$ be two independent discrete random variables. The probability mass function of $X$ is $P(X=k) = \alpha \left[ \frac{k^3}{3} \right]$ for $k \in \{1, 2, 3, 4\}$, and the probability mass function of $Y$ is $P(Y=k) = \beta \left( k^2 + \left[ \frac{k}{2} \right] \right)$ for $k \in \{1, 2, 3, 4, 5\}$, where $[ \cdot ]$ denotes the greatest integer function. Here, $\alpha$ and $\beta$ are constants such that the sum of probabilities for each distribution is $1$. Two subgroups $A$ and $B$ have a number of elements given by $N_A = 256 E[X]$ and $N_B = 366 E[Y]$, where $E[\cdot]$ denotes the expected value. If the mean of subgroup $A$ is $321$ and the mean of subgroup $B$ is $20$, what is the combined mean of the two subgroups?

A dataset consists of an infinite number of subgroups indexed by $n \in \mathbb{N}$. The weight of the $n$-th subgroup is given by $N_n = \left| \alpha^n \right|$, where $\alpha = \int_{0}^{1} x \ln x \, dx$. The mean of the $n$-th subgroup is $\bar{x}_n = \beta^n$, where $\beta = \int_{0}^{\infty} x e^{-2x} \, dx$. What is the combined mean of the entire dataset?

Let $M = \begin{pmatrix} \bar{x}_1 & n_1 \\ n_2 & \bar{x}_2 \end{pmatrix}$ be the parameter matrix for two subgroups $G_1$ and $G_2$, where $n_i$ and $\bar{x}_i$ represent the number of observations and the arithmetic mean of subgroup $G_i$ ($i=1,2$), respectively. Consider the matrices $A = \begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$. If the matrix transformation $T(M) = B M A$ yields $T(M) = \begin{pmatrix} 255 & 445 \\ 180 & 315 \end{pmatrix}$, find the combined arithmetic mean of the dataset formed by merging $G_1$ and $G_2$.

A population $S_n$ is divided into subgroups indexed by triplets of non-negative integers $(r, s, t)$ such that $r + s + t = n$. The weight of the subgroup $(r, s, t)$ is given by $W_{r,s,t} = \frac{n!}{r!s!t!} \left(\frac{1}{n}\right)^s \left(\frac{\lambda}{n^2}\right)^t$, where $\lambda > 0$. The mean of the subgroup $(r, s, t)$ is $X_{r,s,t} = s + 2t$. Let $M_n$ be the combined mean of the population $S_n$. If $\lim_{n \to \infty} (M_n)^n = e^7$, then the value of $\lambda$ is:

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuously differentiable function such that $f(2) = 1$, $f'(2) = -1$, $f(4) = 2$, and $f'(4) = 0$. A dataset consists of two subgroups, $A$ and $B$, with constant sizes $n_A = 2$ and $n_B = 3$ respectively. The mean of subgroup $A$ at time $t$ is given by $\mu_A(t) = \int_{1}^{t} (f(x) + xf'(x)) \,dx$, and the mean of subgroup $B$ at time $t$ is given by $\mu_B(t) = \int_{t}^{t^2} (f(x) + xf'(x)) \,dx$. If $\mu(t)$ represents the combined mean of the entire dataset at time $t$, find the value of $\frac{d\mu}{dt}$ at $t = 2$.

Let $x_1, x_2, x_3, x_4$ be real numbers satisfying the following conditions: $\sum_{i=1}^4 \cos x_i = 0, \quad \sum_{i=1}^4 \sin x_i = 0, \quad \sum_{i=1}^4 x_i = 2\pi$ Find the minimum possible value of the objective function $f(x) = \sum_{i=1}^4 x_i^2$.

Let $\vec{x} = (x_1, x_2, \dots, x_{10}) \in \mathbb{R}^{10}$ be a vector representing a dataset of 10 observations. Let $\vec{u} = (-1, -1, \dots, -1) \in \mathbb{R}^{10}$. The vector $\vec{x}$ is projected onto the line spanned by $\vec{u}$ to obtain the vector $\vec{p}$. It is given that $\|\vec{x} - \vec{p}\|^2 = 90$ and the dot product $\vec{x} \cdot \vec{u} = 50$. If a new dataset is defined as $y_i = 3x_i - 2$ for $i = 1, 2, \dots, 10$, find the value of $\frac{1}{10} \sum_{i=1}^{10} y_i^2$.

Let $A$ and $B$ be two events in a sample space such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$, and $P(A \mid B) = \frac{1}{4}$. For any event $E$, let $I_E$ denote its indicator random variable, which equals $1$ if $E$ occurs and $0$ otherwise. Define a random variable $X$ as: $X = 8 I_A + 8 I_B + 8 I_{A^c \cup B^c} - 2024$ Find the variance of $X$, denoted by $\text{Var}(X)$.

Let $X$ be a discrete random variable with the probability mass function $P(X=x) = \frac{x}{15}$ for $x \in \{1, 2, 3, 4, 5\}$. If $Y = |X - 3|$, what is the ratio of the standard deviation of $Y$ to its expectation, $\frac{\sigma_Y}{E[Y]}$?

Let $f(x) = \begin{vmatrix} 2x+1 & x+2 & x \\ x-1 & 2x-1 & 2x \\ 5x+1 & 4x+3 & 4x+1 \end{vmatrix}$. A data set $S$ consists of the values $y_n = f'(n)$ for $n = 1, 2, \dots, 25$. If the mean and variance of the data set $S$ are $\mu$ and $\sigma^2$ respectively, find the value of $\mu + \sigma^2$.

Let $A = \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}$ be a real orthogonal matrix. Let $S$ be the data set consisting of the 9 entries of $A$. If $\sigma^2$ and $\mu$ are the variance and mean of $S$ respectively, and $V = \frac{\sigma}{|\mu|}$, which of the following statements is true?

A real number $x$ is chosen uniformly at random from the continuous interval $[0, 7]$. Let the random variable $Y$ be defined as $Y = 10 + \int_0^{[x]} (2t - 3) dt$ where $[ \cdot ]$ denotes the greatest integer function. What is the coefficient of variation of $Y$?

A line $L$ with direction cosines $l, m, n$ passes through the point $A(1, -1, 2)$. If $L$ intersects the plane $2x + 2y + 2z = 9$ at a point $B$ such that the distance $AB = \frac{5}{2}$, then the standard deviation of the data set $\{l, m, n\}$ is:

A grouped data distribution has three classes with class marks $x_1 = 1$, $x_2 = 2$, and $x_3 = 3$, and corresponding frequencies $f_1, f_2, f_3$. The total frequency is $\sum_{i=1}^3 f_i = 266$. It is given that the harmonic mean of the distribution is exactly $2$. If $V$ is the maximum possible variance of this distribution, what is the value of $\frac{133^2}{240} V$?

Consider a frequency distribution for a discrete variate $X$, which takes values $x_r = 2|r - 10|$ with corresponding frequencies $f_r = \binom{20}{r}$ for $r = 0, 1, 2, \dots, 20$. Let $P = \frac{1}{2^{20}} \binom{20}{10}$. If the variance of $X$ is $V$, then $V$ is equal to:

A student calculates the mean and variance of 10 real observations $x_1, x_2, \dots, x_{10}$, where $x_k = \text{Re}(z_k) + \text{Im}(z_k)$ and $z_1, z_2, \dots, z_{10}$ are the distinct roots of the equation $z^{10} - 1 = 0$. Later, it was discovered that the student misread the observation corresponding to the root $z = 1$ as $11$ instead of its correct value. What is the incorrect variance calculated by the student?

A student uses the step-deviation method $u_i = \frac{x_i - a}{h}$ with assumed mean $a = 10$ and scale $h = \frac{1}{\sqrt{3} + \sqrt{2}}$ to find the variance of $40$ observations $x_1, x_2, \dots, x_{40}$. The student calculates $\sum_{i=1}^{40} u_i = 40(\sqrt{3} + \sqrt{2} + 1)$ and $\sum_{i=1}^{40} u_i^2 = 2760 + 400\sqrt{6}$. Later, it was discovered that an observation $x_k = 10 + 42(\sqrt{3} - \sqrt{2})$ was incorrectly recorded. If the correct observation is $x_k = 10 + 2(\sqrt{3} - \sqrt{2})$, what is the correct variance of the original data $x$?

Let $S$ be a dataset consisting of $n+1$ observations $x_r = \binom{n}{r}$ for $r = 0, 1, 2, \dots, n$. It is given that the maximum value in $S$ is not unique, and in the binomial expansion of $(1+x)^n$ at $x = \frac{3}{4}$, the numerically greatest term is the $4^{\text{th}}$ term. A student calculates the mean and variance of the dataset $S$. While doing so, they misread exactly one observation of value $n$ as another integer $M$. If the incorrectly calculated mean is $17$, what is the incorrectly calculated variance?

A student calculates the variance of a discrete random variable $X \in \{10, 20, 30\}$ by incorrectly assuming a uniform probability mass function. Later, it is found that the true PMF assigns equal probabilities to $X=10$ and $X=20$, but a different probability to $X=30$. If the true expected value of $X$ is $\frac{369}{15}$, what is the true variance of $X$?

Let $f(x)$ be a continuous probability density function defined on $[0, \infty)$. It is given that $y = f(x)$ satisfies the differential equation $\frac{dy}{dx} + y = \frac{1}{6}(3x^2 - 4x + 2) e^{-x}$ with the initial condition $f(0) = \frac{1}{3}$. If $M$ is the mode of the distribution, find the mean absolute deviation of the distribution about $M$.

Let $c$ be the minimum value of the quadratic expression $Q(t) = \frac{1}{2}t^2 - 4t + 20$. A continuous frequency distribution has exactly three classes of equal width $c$, with the lowest class interval starting at $0$. Let $S$ be the set of all real roots of the equation $(x^2 + x)(x^3 - (e+e^2+e^3)x^2 + (e^3+e^4+e^5)x - e^6) = 0$ The frequencies $f_1, f_2, f_3$ corresponding to the three classes (in increasing order of the class intervals) are the valid, real values of $\ln(\alpha)$ for $\alpha \in S$, sorted in strictly increasing order. Find the mean deviation about the mean for this frequency distribution.

Let $X$ be a discrete random variable taking values in the set $\{1, 2, 3, 4, 5\}$ with probability distribution $P(X=x) = \frac{x}{15}$. If a new random variable is defined as $Y = |X - 3|$, what is the coefficient of variation (defined as the ratio $\frac{\sigma_Y}{\mu_Y}$) of $Y$?

Let $S = \{x, y, z\}$ be a data set of real numbers. A matrix $A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}$ is constructed such that $A$ is an orthogonal matrix with $\det(A) > 0$. Let $\sigma^2$ be the variance of $S$. A student ignores the orthogonality condition and calculates the maximum possible variance of $S$, denoted as $V_0$, under the assumption that $A$ is a singular matrix satisfying $\text{Tr}(A A^T) = 3$. What is the value of $9(\sigma^2 + V_0)$?

Let $S$ be the region in the Argand plane defined by $|z - c| + |z + c| \le D$, where $c$ and $D$ are positive real constants with $D > 2c$. Let $d$ be the minimum distance between any two parallel lines that completely enclose $S$. Let $R = d/2$. Consider the set $X$ consisting of the real parts of the roots of the equation $z^8 - R^8 = 0$. What is the mean deviation of the elements of $X$ about their median?

Let $A$ be a $3 \times 3$ real matrix with distinct eigenvalues $\lambda_1 < \lambda_2 < \lambda_3$ forming an arithmetic progression. A dataset $S$ is constructed by evaluating the function $f(x) = \text{tr}(\text{adj}(xI - A))$ at $x \in \{\lambda_1, \lambda_2, \lambda_3\}$. Let $\mu$ and $M_e$ be the mean and median of $S$, respectively. If $MD_{\mu}$ and $MD_{M_e}$ denote the mean deviation of $S$ about its mean and median, which of the following is INCORRECT?

Let $A$ be a $3 \times 3$ real matrix with three distinct real eigenvalues in an arithmetic progression. Let $S$ be a dataset consisting of the values of the function $f(x) = \text{tr}(\text{adj}(xI - A))$ evaluated at each of the three eigenvalues of $A$. If $MD_{\text{mean}}$ and $MD_{\text{median}}$ denote the mean deviation about the mean and the mean deviation about the median of the dataset $S$ respectively, which of the following expressions identically equals $MD_{\text{mean}} - MD_{\text{median}}$?

A student is asked to calculate the variance of the eigenvalues of a $3 \times 3$ singular matrix $A$. Due to a reading error in a single entry, the student instead uses a non-singular matrix $B$. This error alters the determinant but leaves the trace and the sum of the principal minors of order $2$ unchanged. The student correctly calculates the mean and variance of the eigenvalues of $B$ to be $4$ and $\frac{32}{3}$ respectively.

Upon discovering the error, the student recalculates the variance for the true matrix $A$. However, seeing that $A$ is singular, the student mistakenly discards the zero eigenvalue and calculates the variance of only the non-zero eigenvalues, obtaining a value $V_{\text{wrong}}$. If $V_{\text{true}}$ is the actual variance of all the eigenvalues of $A$, find the value of $|V_{\text{true}} - V_{\text{wrong}}|$.

Let $X$ be a discrete random variable taking values in the set $S = \{3, 6, 9\}$. A student incorrectly assumes that $X$ is uniformly distributed over $S$. Later, the true probability mass function is found to be $P(X=x) = k(-5x^2 + 57x - 72)$ for $x \in S$, where $k$ is a constant. Let $\sigma^2$ be the true variance of $X$. The value of $100 \sigma^2$ is:

Let $x_1, x_2, \dots, x_n$ be $n$ real observations with mean $\bar{x}$ and variance $\sigma^2$. A new variable $u_i$ is defined as $u_i = \frac{x_i - [\bar{x}]}{h}$ for $i = 1, 2, \dots, n$, where $h > 0$ and $[\cdot]$ denotes the greatest integer function. If $\{\cdot\}$ denotes the fractional part function, which of the following represents the sum of squares of the new observations, $\sum_{i=1}^n u_i^2$?

Let $V(x_1, x_2, x_3)$ be the variance of the real numbers $x_1, x_2, x_3$. Let $S$ be the set of all points $(x_1, x_2, x_3) \in \mathbb{R}^3$ such that $\sum_{i=1}^3 \sin x_i = 0$ and $\sum_{i=1}^3 \cos x_i = 0$. Let $M$ be the minimum value of $V$ on $S$. At any point in $S$ where $V$ attains this minimum $M$, the gradient of $V$ can be expressed as $\nabla V = \lambda_1 \nabla \left(\sum_{i=1}^3 \sin x_i\right) + \lambda_2 \nabla \left(\sum_{i=1}^3 \cos x_i\right)$. The value of $\frac{M}{\lambda_1^2 + \lambda_2^2}$ is:

Let $\vec{x} = (x_1, x_2, \dots, x_n)^T \in \mathbb{R}^n$ represent a dataset of $n$ observations with mean $\mu \neq 0$ and variance $\sigma^2 \neq 0$. Let $\vec{e} = (1, 1, \dots, 1)^T \in \mathbb{R}^n$. A new dataset is represented by the vector $\vec{y} = \vec{e} - 2 \left( \frac{\vec{e} \cdot \vec{x}}{\|\vec{x}\|^2} \right) \vec{x}$. Which of the following represents the variance of the dataset $\vec{y}$?

Let $A$ be the set of all real roots of the equation: $(x-2) \sum_{k=1}^{5} k (x-1)^{k-1} = 5(x-1)^5 - (x-1)^4 - 2(x-1)^3 + 5(x-1)^2 - 12(x-1) + 5$ Let $B = \{x \in \mathbb{R} : \sin(\pi x) = 0 \text{ and } 0 \le x \le 5\}$. Let $D$ be the dataset consisting of the values of the function $g(x) = x^2 - x$ evaluated at each element of the set $A \cup (A \cap B)$. What is the coefficient of range of the dataset $D$?

Let $A(\theta) = \begin{pmatrix} \sin\theta & -1 & \cos\theta \\ 1 & \sin\theta & -1 \\ -\cos\theta & 1 & \sin\theta \end{pmatrix}$. If $S$ is the set of all values of $\theta \in [0, 2\pi]$ for which $A(\theta)$ is invertible, find the range of $|A^{-1}(\theta)|$ for $\theta \in S$.

Let $a_1 < a_2 < \dots < a_{2m+1}$ (where $m \in \mathbb{Z}, m \ge 2$) be an arithmetic progression of real numbers with common difference $d > 0$. Let $\mu$ be the arithmetic mean of these numbers. A function $f: \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = \frac{1}{2m+1} \sum_{i=1}^{2m+1} |x - a_i|$. For which of the following intervals is Lagrange's Mean Value Theorem strictly applicable to $f(x)$?

Consider the function $f(x) = \frac{1}{5} \sum_{k=1}^5 |x - k|$. \ \textbf{Statement I:} There does not exist any $c \in (0, 6)$ such that $f'(c) = 0$. \ \textbf{Statement II:} Rolle's Theorem is not applicable to $f(x)$ on the interval $[0, 6]$ because $f(x)$ is non-differentiable at $x = 1, 2, 3, 4, 5$.

Consider two subgroups $A$ and $B$ with sizes $N_A = 9 E[X_A]$ and $N_B = 5 E[X_B]$, where $X_A$ and $X_B$ are discrete random variables. The probability mass function (PMF) of $X_A$ is $P(X_A = k) = c_A \int_{k}^{k+2} \left[ \frac{x}{2} \right] dx$ for $k \in \{1, 3, 5\}$, and the PMF of $X_B$ is $P(X_B = k) = c_B \int_{0}^{k} \{x\} dx$ for $k \in \{1, 2, 3, 4\}$. Here, $[ \cdot ]$ and $\{ \cdot \}$ denote the greatest integer and fractional part functions, respectively. The mean of subgroup $A$ is $366$ and the mean of subgroup $B$ is $256$. Four quantities are defined: $\mu_{comb}$ is the combined mean of $A$ and $B$; $\sigma^2$ is the variance of $X_B$; $\Delta N = N_A - N_B$; and $E = 9 E[X_A]$. Which of the following values is INCORRECTLY matched with its quantity?

An infinite dataset is partitioned into subgroups $G_i$ for $i \in \mathbb{N}$. The weight of each subgroup $G_i$ is given by $W_i = c \left| \frac{2 \cos(i\pi)}{3} \right|^i$, where $c > 0$ is a constant. For each subgroup $G_i$, the arithmetic mean is $\bar{x}_i = \left(-\frac{3}{4}\right)^i$, the variance is $\sigma_i^2 = \frac{i^3 + 2i}{i+1}$, and the median is $M_i = \frac{1}{i}$. What is the combined arithmetic mean of the entire dataset?

Let $p(x)$ be the probability density function of a continuous random variable $X$ defined on the interval $[0, a]$. It is given that $p(x)$ satisfies the differential equation $x p'(x) - p(x) = \frac{6}{5a^3} x^2$ The variance of $X$ is: