The random variable $X$ takes values in $\{-1, 0, 1\}$ with probabilities $P(X = -1) = P(X = 1)$ and $\alpha$ and $P(X = 0) = 1 - 2\alpha$, where $0 < \alpha < \frac{1}{2}$. Let $g(\alpha)$ denote the entropy of $X$ (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?
The problem involves calculating the entropy of a random variable \(X\) with a given probability distribution. Entropy measures the uncertainty or randomness of a random variable. The random variable \(X\) can take values in \(\{-1, 0, 1\}\) with probabilities \(P(X = -1) = \alpha\), \(P(X = 1) = \alpha\), and \(P(X = 0) = 1 - 2\alpha\).
The entropy \(g(\alpha)\) is calculated using the formula:
\(g(\alpha) = - \sum P(x) \log_2 P(x)\)
Breaking it down for this problem:
- Entropy for \(X = -1\): \(-\alpha \log_2(\alpha)\)
- Entropy for \(X = 0\): \(-(1 - 2\alpha) \log_2(1 - 2\alpha)\)
- Entropy for \(X = 1\): \(-\alpha \log_2(\alpha)\)
Thus, the total entropy \(g(\alpha)\) is:
\(g(\alpha) = -2\alpha \log_2(\alpha) - (1 - 2\alpha) \log_2(1 - 2\alpha)\)
Now, let's evaluate the given options using this formula:
- Check for \(g(0.3) \gt g(0.4)\):
Calculate \(g(0.3)\):
\(g(0.3) = -2(0.3) \log_2(0.3) - (1 - 0.6) \log_2(0.4)\)
\(g(0.3) \approx 1.4855\)
Calculate \(g(0.4)\):
\(g(0.4) = -2(0.4) \log_2(0.4) - (1 - 0.8) \log_2(0.2)\)
\(g(0.4) \approx 1.3219\)
Therefore, \(g(0.3) \gt g(0.4)\).
- Check for \(g(0.3) \gt g(0.25)\):
Previously calculated \(g(0.3) \approx 1.4855\).
Calculate \(g(0.25)\):
\(g(0.25) = -2(0.25) \log_2(0.25) - (1 - 0.5) \log_2(0.5)\)
\(g(0.25) \approx 1.5\)
Therefore, \(g(0.3) \gt g(0.25)\).
Therefore, the correct answers are
$g(0.3) > g(0.4)$
and
$g(0.3) > g(0.25)$
.
Important Questions from Information Theory
For any binary (n, h) linear code with minimum distance (2t + 1) or greater \(n - h \ge {\log _2}\left[ {\mathop \sum \limits_{i = 0}^α \left( {\begin{array}{*{20}{c}} n\\ i \end{array}} \right)} \right]\) where α is:
The main processing functions of information system are given below. Arrange them in sequencing order:
(A) Process transaction
(B) Maintain master file
(C) Process enquiry
(D) Process report
(E) Process interactive supper applications
Choose the correct answer from the options given below:
A source transmits symbols from an alphabet of size 16. The value of maximum achievable entropy (in bits) is _______
An analog baseband signal, bandlimited to 100 Hz, is sampled at the Nyquist rate. The samples are quantized into four message symbols that occur independently with probabilities $p_1 = p_4 = 0.125$ and $p_2 = p_3$. The information rate (bits/sec) of the message source is ____________
- Consider a discrete memoryless source with an alphabet of four source symbols.
$s(t)$ is a multi-level (-1, 0, +1, +2) signal representing a long sequence of random symbols from the above source which is generating $10^4$ symbols per second.
Which of the following options is the correct value of equivalent Nyquist bandwidth of $s(t)$?
