Binomial Variance Calculation
The variance of a random variable $X$ following a binomial distribution, denoted as $X \sim \text{Binomial}(n, p)$, is given by the formula:
$ \text{Var}(X) = n p (1-p) $
In this problem, we are given:
- The number of trials, $n = 16$.
- The probability of success, $p = \frac{1}{2}$.
First, calculate the probability of failure, $1-p$:
$ 1-p = 1 - \frac{1}{2} = \frac{1}{2} $
Now, substitute the values of $n$ and $p$ into the variance formula:
$ \text{Var}(X) = 16 \times \frac{1}{2} \times \frac{1}{2} $
$ \text{Var}(X) = 16 \times \frac{1}{4} $
$ \text{Var}(X) = 4 $
Therefore, the variance of the random variable $X$ is 4.
Important Questions from Discrete Distributions
The value of a and b so that the following is probability mass function
X: 0 1 2 P(X = x): 3a 3b 4b with mean 1.1, is:
Digital data received from a sensor can fill up 0 to 32 buffers. Let the sample space be
S = {0, 1, 2, .........., 32} where the sample j denote that j of the buffers are full and \(p\left( i \right) = \frac{1}{{561}}\left( {33 - i} \right)\)
. Let A denote the event that the even number of buffers are full. Then p(A) is :If X is a Poisson random variate with mean 3, then P(|X- 3| < 1) will be:
Let x ∼ N(μ, σ2) If μ2 = σ2, (μ > 0), then the value of P(X < -μ | X < μ) in terms of cumulative function N (0, 1) is:
Consider a binomial random variable X. If X1, X2,...Xn are independent and identically distributed samples from the distribution of X with sum \(Y = \mathop \sum \limits_{i = 1}^n {X_i}\) then the distribution of Y as n → ∞ can be approximated as.
