JNTU
JNTU B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (May - 2019 - R16)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks and may have a, b, c as sub questions.
PART - A
Find the sample space for the experiment of tossing a coin twice.
Answer
The sample space S for the experiment of tossing a coin twice consists of all possible outcomes, where H represents heads and T represents tails. The sample space is defined as:
Consider the experiment of throwing a fair die. Let X be the random variable (r.v.) which assigns 1 if the number that appears is even and 0 if the number that appears is odd. What is the range of X?
Answer
In the experiment of throwing a fair die, the sample space is S = {1, 2, 3, 4, 5, 6}. The random variable X is defined as follows:
- If the outcome is 2, 4, or 6 (even), X = 1.
- If the outcome is 1, 3, or 5 (odd), X = 0.
The range of the random variable X is the set of all possible values it can take, which is {0, 1}.
What is meant by Density functions give an example?
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Define Characteristic Function.
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Write any two properties of Joint Distribution.
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Consider an experiment of tossing a fair coin twice. Let (X, Y) be a bivariate r.v., where X is the number of heads that occurs in the two tosses and Y is the number of tails that occurs in the two tosses. Find P(X = 2, Y = 0)
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Define Stochastic Process give suitable example.
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Define Poisson Random Process and give suitable example.
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Recall any two properties of Power Spectrum.
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What is meant by Cross-Correlation Function?
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PART - B
A family has two children. It is known that at least one of the children is a girl. What is the probability that both the children are girls?
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In a binary communication system a zero and a one is transmitted with probability 0.6 and 0.4 respectively. Due to error in the communication system a zero becomes a one with a probability 0.1 and a one becomes a zero with a probability 0.08. Determine the probability (i) of receiving a one and (ii) that a one was transmitted when the received message is one.
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A single card is drawn from a 52 card deck i) What is the probability that the card is a Jack ii) What is the probability that the card is red 10?
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A company Sells high fidelity amplifiers capable of generating 10, 25 and 50W of audio power. It has on hand 100 of the 10W units of which 15% are defective, 70 of the 25W units with 10% defective, and 30 of the 50W units are 10% defective. i) What is the probability that an amplifier sold from the 10W units is defective. ii) What is the probability that a unit randomly selected for a sale is defective.
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In a binary communication system, the probability of bit error is 0.01. If a block of 8 bits are transmitted, find the probability that i) Exactly 2 bit errors will occur ii) At least 2 bit errors will occur
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Consider the Random variable X with the Probability Mass function (pmf) as follows. Estimate its Mean and Variance.
| Value of the Random Variable X | 0 | 1 | 2 | 3 |
| Px(X) | 1/8 | 1/8 | 1/4 | 1/2 |
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Suppose the waiting time of packets is X in computer Network is an exponential r.v. with
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Suppose X is a random variable defined by the pdf
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The joint pdf of two random variables X and Y are given by
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Suppose X and Y are two jointly-Gaussian 0-mean random variables with variances of 1 and 4 respectively and a covariance of 1. Estimate the joint PDF f(x,y).
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Two statistically independent r.v. X and Y have Mean Values E[X]=2 and E[Y]=4. They have second Moments E[X^2]=8 and E[Y^2]=25. Determine: i) The Mean Value ii) The second moment and iii) The Variance of the random Variable W = 3X – Y
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Explain about the concept of Stationarity and Statistical Independence.
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Explain about Autocorrelation Function and its Properties.
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Explain about Cross-Correlation Functions and Gaussian Random Processes.
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Explain about the Stochastic Process Concept, Classification of Processes.
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State and Prove the relation between Power spectrum density and Auto correlation Function.
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State and Prove the Relationship between Cross-Power Spectrum and Cross-Correlation Function.
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