We are given that the random variable \( X \) follows a binomial distribution \( X \sim B(n, p) \) with the following properties: The mean is \( E(X) = np = 1 \). The probability \( P(X = 2) = \frac{27}{128} \).
Step 1: Determine \( p \) using the Mean
Since the mean of a binomial distribution is \( E(X) = np \), we have: \[ np = 1 \quad \Rightarrow \quad p = \frac{1}{n}. \]
Step 2: Use \( P(X = 2) \)
For a binomial distribution, the probability of exactly 2 successes is: \[ P(X = 2) = \binom{n}{2} p^2 (1-p)^{n-2}. \] Substitute \( p = \frac{1}{n} \) and the given \( P(X = 2) = \frac{27}{128} \): \[ \binom{n}{2} \left(\frac{1}{n}\right)^2 \left(1 - \frac{1}{n}\right)^{n-2} = \frac{27}{128}. \] Using the binomial coefficient \( \binom{n}{2} = \frac{n(n-1)}{2} \), the equation becomes: \[ \frac{n(n-1)}{2} \cdot \frac{1}{n^2} \cdot \left(1 - \frac{1}{n}\right)^{n-2} = \frac{27}{128}. \] Simplify to obtain: \[ \frac{(n-1)}{2n} \cdot \left(1 - \frac{1}{n}\right)^{n-2} = \frac{27}{128}. \]
Step 3: Solve for \( n \)
Due to the complexity of the expression, we test possible values for \( n \). By substituting \( n = 4 \), we find that: \[ P(X = 2) = \frac{27}{128}, \] confirming that \( n = 4 \) satisfies the condition.
Step 4: Compute the Variance
The variance of a binomial distribution is given by: \[ \text{Var}(X) = np(1-p). \] Using \( n = 4 \) and \( p = \frac{1}{4} \) (since \( p = \frac{1}{n} \)), we have: \[ \text{Var}(X) = 4 \times \frac{1}{4} \times \left(1 - \frac{1}{4}\right) = 1 \times \frac{3}{4} = \frac{3}{4}. \] Thus, the variance of \( X \) is: \[ \boxed{\frac{3}{4}}. \]
If the coefficient of \( x^r \) in the expansion of \( (1 + x + x^2)^{100} \) is \( a_r \), and \( S = \sum\limits_{r=0}^{300} a_r \), then
\[ \sum\limits_{r=0}^{300} r a_r = \]
Given below are two statements, one is labelled as Assertion (A) and the other one labelled as Reason (R).
Assertion (A): \[ 1 + \frac{2.1}{3.2} + \frac{2.5.1}{3.6.4} + \frac{2.5.8.1}{3.6.9.8} + \dots \infty = \sqrt{4} \] Reason (R): \[ |x| <1, \quad (1 - x)^{-1} = 1 + nx + \frac{n(n+1)}{1.2} x^2 + \frac{n(n+1)(n+2)}{1.2.3} x^3 + \dots \]
\[ \text{The domain of the real-valued function } f(x) = \sin^{-1} \left( \log_2 \left( \frac{x^2}{2} \right) \right) \text{ is} \]
Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix with positive integers as its elements. The elements of \( A \) are such that the sum of all the elements of each row is equal to 6, and \( a_{22} = 2 \).
\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]