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
- Match items in List – I with items in List – II and select the correct answer from the code given below :
List – I List – II a. The Screening Hypothesis 1. George A. Akerlof b. Job Market Signalling 2. K.J. Arrow c. The problem of moral hazard in the case of medical insurance 3. M. Spencer d. The market for lemons 4. Paul W. Miller and Paul A. Volcker
Codes : 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 $s(t)$? An information system uses the following daily workflow. Each function can start only after the input it explicitly requires has been produced. Arrange the functions in execution order.
(A) Process and validate the day’s sales transactions.
(B) Update the inventory master file using the validated transactions.
(C) Process managers’ enquiries using the figures in the completed daily report.
(D) Generate the daily report from the updated inventory master file.
(E) Run interactive decision-support scenarios using the results returned by those enquiries.
Choose the correct answer from the options below.
