Random Signals And Noise Practice Questions

Let $U$ and $V$ be two independent zero mean Gaussian random variables of variances $\frac{1}{4}$ and $\frac{1}{9}$ respectively. The probability $P\left ( 3V\geq 2U \right )$ is

Consider two identically distributed zero-mean random variables U and V. Let the cumulative distribution functions of U and 2V be $F(x)$ and $G(x)$ respectively. Then, for all values of $x$

The power spectral density of a real process X(t) for positive frequencies is shown below. The values of $E[X^{2}(t)]$ and |E[X(t)]|, respectively, are

Two independent random variables X and Y are uniformly distributed in the interval [-1,1]. The probability that max[X,Y] is less than 1/2 is

X(t) is a stationary random process with auto-correlation function $R_{x}(\tau )=exp(-\pi \tau^{2})$ . This process is passed through the system shown below. The power spectral density of the output process Y(t) is

Consider a baseband binary PAM receiver shown below. The additive channel noise n(t) is with power spectral density $S_{N}(f)=N_{0}/2=10^{-20}W/Hz$ . The lowpass filter is ideal with unity gain and cut-off frequency 1 MHz. Let $Y_{k}$ represent the random variable y( $t_{k}$ ). $Y_{k}$ = $N_{k}$ if transmitted bit $b_{k}=0$ $Y_{k}=a+ N_{k}$ if transmitted bit $b_{k}=1$ Where $N_{k}$ represents the noise sample value. The noise sample has a probability density function, $P_{Nk}(n)=0.5\alpha e^{-\alpha |n|}$ (This has mean zero and variance 2/ $\alpha ^{2}$ ). Assume transmitted bits to be equiprobable and threshold z is set to a/2 = $10^{-6}$ V. The probability of bit error is

X(t) is a stationary process with the power spectral density $S_{x}(f) \gt 0$ , for all f. The process is passed through a system shown below Let $S_{y}(f)$ be the power spectral density of Y(t). Which one of the following statements is correct

A white noise process X(t) with two-sided power spectral density $1 \times 10^{-10}W/Hz$ is input to a filter whose magnitude squared response is shown below. The power of the output process Y(t) is given by

If the power spectral density of stationary random process is a sine-squared function of frequency, the shape of its auto correlation is

A discrete random variable X takes values from 1 to 5 with probabilities as shown in the table. A student calculates the mean X as 3.5 and her teacher calculates the variance of X as 1.5. Which of the following statements is true ?

Noise with double-sided power spectral density on K over all frequencies is passed through a RC low pass filter with 3 dB cut-off frequency of $f_{c}$ . The noise power at the filter output is

The probability density function (pdf) of random variable X is as shown below The corresponding commutative 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 real numbers. The equation relating M and N is

If $R(\tau)$ is the auto correlation function of a real, wide-sense stationary random process, then which of the following is NOT true

If S(f) is the power spectral density of a real, wide-sense stationary random process, then which of the following is ALWAYS true?

If E denotes expectation, the variance of a random variable X is given by

A zero-mean white Gaussian noise is passes through an ideal lowpass filter of bandwidth 10 kHz. The output is then uniformly sampled with sampling period $t_{s}$ = 0.03 msec. The samples so obtained would be

The following question refer to wide sense stationary stochastic processes It is desired to generate a stochastic process (as voltage process) with power spectral density S( $\omega$ ) = 16/(16 + $\omega ^{2}$ ) by driving a Linear-Time-Invariant system by zero mean white noise (As voltage process) with power spectral density being constant equal to 1. The system which can perform the desired task could be

A uniformly distributed random variable X with probability density function $f_{x}(x)=\frac{1}{10}(u(x+5)-u(x-5))$ where u(.) is the unit step function is passed through a transformation given in the figure below. The probability density function of the transformed random variable y would be ,