# Random Variables MCQs For ESE | Random Process and Noise MCQs

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## Random Variables MCQs For ESE

Q.1 A system has a receiver noise resistance of 50\Omega. it is connected to an antenna with an input resistance of 50\Omega. The noise figure of the system is

• (a) 1
• (b) 2
• (c) 50
• (d) 101

Explanation:

Ra \rightarrow Resistance of antenna

F = 1 + \frac{R_{eq}}{R_a} =1 + \frac{50}{50} = 2

Q.2 Which one of the following types of noise gains importance at high frequency?

• (a) Shot noise
• (b) Random noise
• (c) Impulse noise
• (d) Transit-time noise

• (a) change in polarization only at the receiver end
• (b) change in frequency only at the receiver end
• (c) fluctuation in signal strength at the receiver
• (d) change in phase only at the receiver end

Q.4 Thermal noise is passed through an ideal low-pass filter having a cutoff at fc = w Hz. The autocorrelation value of the noise at the output of the filter is given as

• (a) A delta function at t = 0
• (b) Gaussian over the range -\infty \leq t \leq \infty
• (c) Sinc function over the range -\infty < t < \infty
• (d) Triangular function over the range -\frac{1}{2w}\leq t \leq \frac{1}{2w}

Answer: (c) Sinc function over the range -\infty < t < \infty

Q.5 A random process obeys Poisson’s distribution. It is given that the mean of the process is 5. Then the variance of the process is

• (a) 5
• (b) 0.5
• (c) 25
• (d) 0

Explanation: We know that standard deviation = S.D

\sqrt{Mean} = \sigma_X = \sqrt{\sigma_x^2}=\sqrt{Variance}

\therefore Variance = mean = 5

For Poisson’s distribution mean and variance are the same.

Q.6 Which one of the following gives the average value or expectation of the function g(x) of the random variable X? {Given f(x) is the probability density function}

(a) E[g(x)] = \int_{-\infty}^{\infty} g(X)dX

(b) E[g(x)] = \int_{-\infty}^{\infty} g(X)f(X)dX

(c) E[g(x)] = \int_{-\infty}^{\infty} g^*(X)dX

(d) E[g(x)] = \int_{-\infty}^{\infty} \left[ \frac{g(X)}{f(X)}\right ]

Answer: (b) E[g(x)] = \int_{-\infty}^{\infty} g(X)f(X)dX

Q.7 The auto-correlation function R_X(\tau) of a random process has the property that R_X(0) is equal to

(a) The square of the mean value of the process

(b) The mean squared value of the process

(c) The smallest value of R_X(\tau)

(d) 1/2 \left[ R_X(\tau) + R_X(-\tau)\right ]

Answer: (b) The mean squared value of the process

Q.8 If a linear time-invariant system is excited by a truly random signal like white noise, the output of the linear system will have which of the following properties

• (a) Output will be a white noise
• (b) Output will be periodic
• (c) Output will not be random
• (d) Output will be correlated or colored noise

Answer: (a) Output will be a white noise

Q.9 Which of the following is/are not a property/properties of a power spectral density function S_X(w)?

• (a) S_X(w) is a real function of w
• (b) S_X(w) is an even function of w
• (c) S_X(w) is a non-positive function of w. i.e. S_X(w) \leq 0 for all w
• (d) All of the above

Answer: (c) S_X(w) is a non-positive function of w. i.e. S_X(w) \leq 0 for all w

Q.10 What is the spectral density of white noise?

• (a) A constant
• (b) \delta(w)
• (c) [\delta(w)]^2
• (d) A step function in w

Q.11 Let x(n) be a real-valued sequence that is a sample sequence of a wide-sense stationary discrete-time random process. The power density function of this signal is

• (a) real, odd, and non-negative
• (b) real, even, and non-negative
• (c) purely imaginary, even, and negative
• (d) purely imaginary, odd, and negative

Answer: (b) real, even, and non-negative

Q.12 Which one of the following is the correct statement? if the value of the resistor creating thermal noise is doubled, the noise generated is

• (a) halved
• (b) doubled
• (c) unchanged
• (d) slightly changed

Q.13 The output of two noise sources each producing uniformly distributed noise over the range -a to +a are added. What is the p.d.f. of the added noise?

• (a) Uniformly distributed over the range -2a to +2a
• (b) Triangular over the range -2a to +2a
• (c) Gaussian over the range -\infty to \infty
• (d) None of the above

Answer: (b) Triangular over the range -2a to +2a

Q.14 If random process x(t) and y(t) are orthogonal then, statistical averages

• (a) and time averages are different
• (b) and time averages are the same
• (c) are greater than time averages
• (d) are smaller than time averages

Answer: (b) and time averages are the same

Q.15 The spectral density and autocorrelation function of white noise is

• (a) Delta and uniform
• (b) Uniform and delta
• (c) Gaussian and uniform
• (d) gaussian and delta

Q.16 The probability cumulative distribution function must be monotone and

• (a) increasing
• (b) decreasing
• (c) non-increasing
• (d) non-decreasing

Q.17 For a random signal (continuous time) x(t) defined for t >0, its probability density function (pdf) at t=t0 is such that

• (a) It is non-negative and its integral equals 1
• (b) Need not be non-negative, but integral equal 1
• (c) It is non-negative, but the integral is not 1
• (d) Non of the above

Answer: (a) It is non-negative and its integral equals 1

Q.18 The noise factor of an attenuator pad that has an insertion loss of 6 dB is

• (a) 0.25
• (b) 0.5
• (c) 2
• (d) 4

Q.19 The cumulative distribution function of a random variable x is the probability that X takes the value.

• (a) less than or equal to x
• (b) equal to x
• (c) greater than x
• (d) Zero

Answer: (a) less than or equal to x

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