Recent questions tagged probability-density-function

A random variable $\mathrm{X}$, distributed normally as $\mathrm{N(0,1)}$ undergoes the transformation$\mathrm{Y}=\mathrm{h}(\mathrm{X})$, given in the figure. The form o...

Suppose that a random variable $X$ has a probability density function\[\begin{aligned}f(x) & =c(x-4) \quad \text { for } 4 \leq x \leq 6 \\& =0 \quad \text { for all othe...

A non-negative loss in a car accident is distributed with the following probability density function\[f(x)=\frac{1}{10} \exp (-x / 10)\]for $x \geq 0$. Suppose that first...

Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[-1,1]$. The probability that $X^{2}+Y^{2}>1$ is$\pi /...

Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. When $\alpha \geq 1$, the probab...

Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. The probability that $\text{X + ...

Suppose you throw a dart and it lands uniformly at random on a target which is a disk of unit radius. What is the probability density function $f(x)$ of the distance of t...

Anu reached a bus stop at $\text{9:00 AM.}$ She knows that the number of minutes after $\text{9:00 AM}$ when the bus will arrive is distributed with probability density f...

Consider a signal $X$ that can take two values, $-1$ with probability $p$ and $+1$ with probability $1-p$. Let $Y=X+N$, where $N$ is mean zero random noise that has proba...

A continuous-time signal with finite energy, band limited from $3 \; \mathrm{MHz}$ to $5 \; \mathrm{MHz}$, is ideally sampled, encoded by a fixed length PCM coder and the...

An output of a communication channel is a random variable $v$ with the probability density function as shown in the figure The mean square value of $v$ is$4$$6$$8$$9$

A uniformly distributed random variable $X$ with probability density function\[ f_{\gamma}(x)=\frac{1}{10}(t r(x+5)-u(x-5)) \]where $u(.)$ is the unit step function is pa...

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...

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

(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...

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...

$X$ is a random variable with uniform probability density function in the interval $[-2,\:10]$. For $Y=2X-6$, the conditional probability $P\left ( Y\leq 7\mid X\geq 5 \r...

Two random variables $X$ and $Y$ are distributed according to $$f_{X,Y}(x,y)=\begin{cases} (x+y),& 0\leq x\leq 1,&0\leq y\leq 1\\ 0, & \text{otherwise.} \end{cases}$$ The...

A source emits bit $0$ with probability $\frac{1}{3}$ and bit $1$ with probability $\frac{2}{3}$. The emitted bits are communicated to the receiver. The receiver decides ...

The input to a $1$ – bit quantizer is a random variable $X$ with pdf $f_{X}( x )= 2e^{-2x}$ for $x\geq 0$ and $f_{X} (x )= 0$ for $x< 0$. For outputs to be of equal proba...

A binary source generates symbols $X\in \left \{ -1,1 \right \}$ which are transmitted over a noisy channel.The probability of transmitting $X=1$ is $0.5.$ Input to the t...