Recent questions tagged probability-and-statistics

An input to a $6$-level quantizer has the probability density function $f(x)$ as shown in the figure. Decision boundaries of the quantizer are chosen so as to maximize th...

An input to a $6$-level quantizer has the probability density function $f(x)$ as shown in the figure. Decision boundaries of the quantizer are chosen so as to maximize th...

The probability density function (PDF) of a random variable $\mathrm{X}$ is as shown below.The corresponding cumulative distribution function (CDF) has the form

$P_{X}(x)=M \exp (-2|x|)+N \exp (-3|x|)$ is the probability density function for the real random variable $X$, over the entire $x$ axis. $M$ and $N$ are both positive rea...

A memoryless source emits $n$ symbols each with a probability $p$. The entropy of the source as a function of $n$increases as $\log n$decreases as $\log (1 / n)$increases...

Consider a Binary Symmetric Channel (BSC) with probability of error being $p$. To transmit a bit, say $1,$ we transmit a sequence of three $1\text{s}.$ The receiver will ...

In radioactive decay, the disintegration rate of the nuclei isconstant at all timesinversely proportional to half-life of the nucleiinversely proportional to the number o...

The function shown in figure, can represent a probability density function for $\mathrm{A} \dots$

A fair coin is tossed $10$ times. What is the probability that $\text{ONLY}$ the first two tosses will yield heads?$\left(\frac{1}{2}\right)^{2}$${ }^{10} C_{2}\left(\fra...

Consider two independent random variables $X$ and $Y$ with identical distributions. The variables $X$ and $Y$ take values $0,1$ and $2$ with probabilities $\frac{1}{2}, \...

A discrete random variable $X$ takes values from $1$ to $5$ with probabilities as shown in the table. A student calculates the mean of $X$ as $3.5$ and her teacher calcul...

The amplitude of a random signal is uniformly distributed between $-5 \mathrm{~V}$ and $5 \mathrm{~V}$.If the positive values of the signal are uniformly quantized with a...

A fair coin is tossed independently four times. The probability of the event "the number of times heads show up is more than the number of times tails show up" is$\frac{1...

Statement for Linked Answer Questions 54 and 55:Consider a baseband binary PAM receiver shown below. The additive channel noise $n(t)$ is white with power spectral densit...

Statement for Linked Answer Questions 54 and 55:Consider a baseband binary PAM receiver shown below. The additive channel noise $n(t)$ is white with power spectral densit...

For a random variable $x$ following the probability density function, $\mathrm{p}(x)$, shown in given figure the mean and the variance are, respectively,$1 / 2$ and $2 / ...

Two resistors $R_1$ and $R_2$ (in ohms) at temperatures $T_1$ and $T_2 K$ respectively, are connected in series. Their equivalent noise temperatures is ___________ $K$.

A binary source has symbol probabilities $0.8$ and $0.2$. If extension coding (blocks of $4$ symbols) is used. The lower and upper bounds on the average code word length ...

(a) A Gaussian random variable with zero mean and variance $\sigma$ is input to a limiter with input output characteristic given by$$ \begin{array}{ll} e_{\text {out }}=e...

A fair dice is tossed two times. The probability that the second toss results in a value that is higher than the first toss is$2 / 36$$2 / 6$$5 / 12$$1 / 2$

A four-phase and an eight-phase signal constellation are shown in the figure below. Assuming high $\text{SNR}$ and that all signals are equally probable, the additional a...

Let $H(X)$ denote the entropy of a discrete random variable $X$ taking $K$ possible distinct real values. Which of the following statements is/are necessarily true?$H(X) ...

A state transition diagram with states $A, B,$ and $C,$ and transition probabilities $p_{1}, p_{2}, \dots, p_{7}$ is shown in the figure (e.g., $\text{p}_{1}$ denotes the...

Consider communication over a memoryless binary symmetric channel using a $(7, 4)$ Hamming code. Each transmitted bit is received correctly with probability $(1 – \in),$ ...

Consider a channel over which either symbol $x_{A}$ or symbol $x_{B}$ is transmitted. Let the output of the channel $Y$ be the input to a maximum likelihood $\text{(ML)}$...

Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, $f(x),$ as shown in the fi...

Two continuous random variables $X$ and $Y$ are related as$$Y=2X+3$$Let $\sigma ^{2}_{X}$ and $\sigma ^{2}_{Y}$denote the variances of $X$ and $Y$, respectively. The vari...

The autocorrelation function $R_{X}\left ( \tau \right )$ of a wide-sense stationary random process $X(t)$ is shown in the figure.The average power of $X(t)$ is _________...

A speech signal, band limited to $4\:\text{kHz}$, is sampled at $1.25$ times the Nyquist rate. The speech samples, assumed to be statistically independent and uniformly d...