If a variable π§ shows a standard normal distribution, then the percent probability that $0 β€ π§^2 β€ 1$ is ________ (rounded off to the nearest integer).
- 34
- 68
- 95
- 99
The Correct Option is B
Solution and Explanation
The standard normal distribution has 68% of its probability within one standard deviation of the mean. Since z2 is bounded between 0 and 1, the percentage probability is 68%.
Top Questions on Normal distribution
Let \( (X, Y)^T \) follow a bivariate normal distribution with \[ E(X) = 2, \, E(Y) = 3, \, {Var}(X) = 16, \, {Var}(Y) = 25, \, {Cov}(X, Y) = 14. \] Then \[ 2\pi \left( \Pr(X>2, Y>3) - \frac{1}{4} \right) \] equals _________ (rounded off to two decimal places).
- GATE ST - 2025
- Statistics
- Normal distribution
- Let \( (X_1, X_2, X_3)^T \) have the following distribution
\[ N_3 \left( \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 & 0.4 & 0 \\ 0.4 & 1 & 0.6 \\ 0 & 0.6 & 1 \end{pmatrix} \right). \]
Then the value of the partial correlation coefficient between \( X_1 \) and \( X_2 \) given \( X_3 \) is __________ (rounded off to two decimal places).- GATE ST - 2025
- Statistics
- Normal distribution
Let \( \{ X_n \}_{n \geq 1} \) be a sequence of independent random variables with \[ \Pr(X_n = -\frac{1}{2^n}) = \Pr(X_n = \frac{1}{2^n}) = \frac{1}{2}, \quad \forall n \in \mathbb{N}. \] Suppose that \( \sum_{i=1}^{n} X_i \) converges to \( U \) as \( n \to \infty \). Then \( 6 \Pr(U \leq \frac{2}{3}) \) equals ___________ (answer in integer).
- GATE ST - 2025
- Statistics
- Normal distribution
For \( Y \in \mathbb{R}^n \), \( X \in \mathbb{R}^{n \times p} \), and \( \beta \in \mathbb{R}^p \), consider a regression model \[ Y = X \beta + \epsilon, \] where \( \epsilon \) has an \( n \)-dimensional multivariate normal distribution with zero mean vector and identity covariance matrix. Let \( I_p \) denote the identity matrix of order \( p \). For \( \lambda>0 \), let \[ \hat{\beta}_n = (X^T X + \lambda I_p)^{-1} X^T Y, \] be an estimator of \( \beta \). Then which of the following options is/are correct?
- GATE ST - 2025
- Statistics
- Normal distribution
Let \( X = (X_1, X_2, X_3)^T \) be a 3-dimensional random vector having multivariate normal distribution with mean vector \( (0, 0, 0)^T \) and covariance matrix
\[ \Sigma = \begin{pmatrix} 4 & 0 & 0 <br> 0 & 9 & 0 <br> 0 & 0 & 4 \end{pmatrix}. \]
{Let } \( \alpha^T = (2, 0, -1) \) { and } \( \beta^T = (1, 1, 1) \).
Then which of the following statements is/are correct?- GATE ST - 2025
- Statistics
- Normal distribution
Questions Asked in IIT JAM BT exam
- A box of mass 20 ππ is pulled at constant speed across a floor by a rope. The rope makes an angle of 45Β° with the horizontal. Assuming that friction is negligible, the work done in pulling the box by a distance of 20 π is _____ π½ (rounded off to the nearest integer). (Use $π = 9.8 ππ ^{β2}$)
- IIT JAM BT - 2024
- Friction
- The value of $\lim_{π₯β3}\frac{(π₯^2 β9)}{(π₯^2β4π₯+3 )} $ is (rounded off to the nearest integer).
- IIT JAM BT - 2024
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- Which one of the following statements about polyploidy is correct?
- IIT JAM BT - 2024
- Genetics
- A ball dropped from a bridge hits the surface of the water in 3 π . The height of the bridge, ignoring air resistance, is ___ π (rounded off to one decimal place). (Use $π = 9.8 ππ ^{β2}$)
- IIT JAM BT - 2024
- Motion in a plane
- The restriction map of a circular plasmid is shown below, along with the indicated distances between the restriction sites.
The plasmid was completely digested with EcoRI, and XhoI. The products were analysed by agarose gel electrophoresis followed by ethidium bromide staining. The number of bands that will be visible in the gel when exposed to UV light is __________.- IIT JAM BT - 2024
- Biotechnology