List of top Mathematics Questions on Probability and Statistics asked in TS PGECET

A random variable \(X\) has the probability density function \[ f(x)=Kx,\qquad 0<x<2, \] then the ratio between \(K\) and the mean of \(X\) is
If variance of \(X\) is \(4\), then the variance of \(Y=4X+2\) is
Suppose that \(X\) and \(Y\) are independent random variables with probability densities \[ f_X(x)= \begin{cases} \dfrac{8}{x^3}, & x>2,[2mm] 0, & \text{elsewhere}, \end{cases} \qquad f_Y(y)= \begin{cases} 2y, & 0 0, & \text{elsewhere}. \end{cases} \] Then the expected value of \(Z=XY\) is
A continuous random variable \(X\) has the density function \[ f(x)= \begin{cases} e^{-x}, & x>0, 0, & \text{elsewhere}, \end{cases} \] then the expected value of \[ g(x)=e^{\frac{2x}{3}} \] is
If \[ F(x)= \begin{cases} 0, & x[2mm] \left(\dfrac{x+7}{a}\right), & -7[2mm] 1, & x>7, \end{cases} \] represents the probability distribution function of a continuous random variable, then \(a=\)
Suppose that \(X\) and \(Y\) are independent random variables with probability densities \[ f_X(x)= \begin{cases} \dfrac{8}{x^3}, & x>2,[2mm] 0, & \text{elsewhere}, \end{cases} \qquad f_Y(y)= \begin{cases} 2y, & 0 0, & \text{elsewhere}. \end{cases} \] Then the expected value of \(Z=XY\) is
A continuous random variable \(X\) has the density function \[ f(x)= \begin{cases} e^{-x}, & x>0, 0, & \text{elsewhere}, \end{cases} \] then the expected value of \[ g(x)=e^{\frac{2x}{3}} \] is
If \[ F(x)= \begin{cases} 0, & x[2mm] \left(\dfrac{x+7}{a}\right), & -7[2mm] 1, & x>7, \end{cases} \] represents the probability distribution function of a continuous random variable, then \(a=\)
If a random variable \( X \), defined such that \( E(X-1)^2 = 10 \) and \( E(X-2)^2 = 6 \) then mean \( (\mu) \) and variance \( (\sigma^2) \) are
If a random variable \( X \), defined such that \( E(X-1)^2 = 10 \) and \( E(X-2)^2 = 6 \) then mean \( (\mu) \) and variance \( (\sigma^2) \) are
The joint probability density function of a two-dimensional random variable \((X,Y)\) is \[ f(x,y)= \begin{cases} 2, & 0\\ 0, & \text{otherwise}. \end{cases} \] Which of the following is correct?