>
Exams
>
Mathematics
>
Probability and Statistics
>
if a random variable x has the following probabili
Question:
If a random variable \( X \) has the following probability distribution, then its variance is nearly:
Show Hint
To find the variance of a probability distribution, compute \( E(X^2) \) and use \( \text{Var}(X) = E(X^2) - (E(X))^2 \).
AP EAMCET - 2024
AP EAMCET
Updated On:
May 20, 2025
\( 2.8875 \)
\( 2.9875 \)
\( 2.7865 \)
\( 2.785 \)
Hide Solution
Verified By Collegedunia
The Correct Option is
A
Solution and Explanation
Step 1: Verify that probabilities sum to 1
We have: \[ 0.05 + 0.1 + 2K + 0 + 0.3 + K + 0.1 = 1. \] Solving for \( K \): \[ 2K + K = 1 - (0.05 + 0.1 + 0.3 + 0.1) = 1 - 0.55 = 0.45. \] \[ 3K = 0.45 \Rightarrow K = 0.15. \]
Step 2: Compute Expectation \( E(X) \)
\[ E(X) = \sum x P(X=x). \] \[ = (-3)(0.05) + (-2)(0.1) + (-1)(2K) + (0)(0) + (1)(0.3) + (2)(K) + (3)(0.1). \] \[ = (-0.15) + (-0.2) + (-0.3) + 0 + 0.3 + 0.3 + 0.3 = 0. \]
Step 3: Compute \( E(X^2) \)
\[ E(X^2) = \sum x^2 P(X=x). \] \[ = (-3)^2(0.05) + (-2)^2(0.1) + (-1)^2(2K) + (0)^2(0) + (1)^2(0.3) + (2)^2(K) + (3)^2(0.1). \] \[ = (9)(0.05) + (4)(0.1) + (1)(0.3) + 0 + (1)(0.3) + (4)(0.15) + (9)(0.1). \] \[ = 0.45 + 0.4 + 0.3 + 0.3 + 0.6 + 0.9 = 2.8875. \]
Step 4: Compute Variance \( \sigma^2(X) \)
\[ \text{Var}(X) = E(X^2) - (E(X))^2. \] Since \( E(X) = 0 \), \[ \text{Var}(X) = 2.8875 - 0^2 = 2.8875. \]
Step 5: Conclusion
Thus, the final answer is: \[ \boxed{2.8875}. \]
Download Solution in PDF
Was this answer helpful?
0
1
Top Questions on Probability and Statistics
Let $X$ be an exponential random variable with mean parameter one. Then the conditional probability $P(X>10 | X>5)$ is equal to
AP PGECET - 2025
Mathematics
Probability and Statistics
View Solution
The number of accidents occurring in AU region in a month follows Poisson distribution with mean $\lambda = 5$. The probability of less than two accidents in a randomly selected month is ______
AP PGECET - 2025
Mathematics
Probability and Statistics
View Solution
A machine produces 0, 1 or 2 defective items in a day with probabilities of \( \frac{1}{4}, \frac{1}{2}, \frac{1}{4} \) respectively. Then, the standard deviation of the number of defective items produced by the machine in a day is ............
AP PGECET - 2025
Mathematics
Probability and Statistics
View Solution
If \( X \) is a continuous random variable with the probability density
\[ f(x) = \begin{cases} K(1 - x^3), & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases} \]
Then, the value of \( K \) is ............
AP PGECET - 2025
Mathematics
Probability and Statistics
View Solution
The probability of a component being defective is 0.01. There are 100 such components in a machine. Then the probability of two or more defective components in the machine is _______
AP PGECET - 2025
Mathematics
Probability and Statistics
View Solution
View More Questions
Questions Asked in AP EAMCET exam
If a real valued function \( f: [a, \infty) \to [b, \infty) \) is defined by \( f(x) = 2x^2 - 3x + 5 \) and is a bijection, then find the value of \( 3a + 2b \):
AP EAMCET - 2024
Functions
View Solution
Let \( \alpha \in \mathbb{R} \). If the line \( (a + 1)x + \alpha y + \alpha = 1 \) passes through a fixed point \( (h, k) \) for all \( a \), then \( h^2 + k^2 = \):
AP EAMCET - 2024
general equation of a line
View Solution
The length of a metal bar is 20 cm and the area of cross-section is \( 4 \times 10^{-4} \) m\(^2\). If one end of the rod is kept in ice at \( 0^\circ C \) and the other end is kept in steam at \( 100^\circ C \), the mass of ice melted in one minute is 5 g. The thermal conductivity of the metal in Wm\(^{-1}\)K\(^{-1}\) is
(Latent heat of fusion = 80 cal/gm)
AP EAMCET - 2024
Energy Conservation
View Solution
An alternating current is given by \( i = (3 \sin \omega t + 4 \cos \omega t) \) A. The rms current will be:
AP EAMCET - 2024
Alternating current
View Solution
A lamp is rated at 240V, 60W. When in use the resistance of the filament of the lamp is 20 times that of the cold filament. The resistance of the lamp when not in use is:
AP EAMCET - 2024
thermal properties of matter
View Solution
View More Questions