JNTU
JNTU B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (March 2022 - R15)
Answer any five questions. All questions carry equal marks.
Main
An aircraft is used to fire at a target. It will be successful if 2 or more bombs hit the target. If the aircraft fires 3 bombs and the probability of the bomb hitting the target is 0.4, then what is the probability that the target is hit?
Answer
Let X be the random variable representing the number of bombs that hit the target. Since each bomb hit is an independent Bernoulli trial with probability p = 0.4 and n = 3, X follows a binomial distribution: X ~ B(n=3, p=0.4).
The probability mass function is given by:
The target is successful if 2 or more bombs hit, which is P(X ≥ 2) = P(X = 2) + P(X = 3).
Calculating each term:
Therefore, the total probability is:
State and prove the total probability theorem.
Answer
The Total Probability Theorem states that if a set of events B1, B2, ..., Bn forms a partition of the sample space S (meaning they are mutually exclusive and their union is S), then for any event A in S, the probability of A is given by:
Proof:
- Since the events Bi partition the sample space S, we can write A as the union of the intersections of A with each Bi: A = A ∩ S = A ∩ (B1 ∪ B2 ∪ ... ∪ Bn).
- By the distributive law of sets, this becomes: A = (A ∩ B1) ∪ (A ∩ B2) ∪ ... ∪ (A ∩ Bn).
- Because the Bi are mutually exclusive, the sets (A ∩ Bi) are also mutually exclusive.
- Using the axiom of additivity for probability: P(A) = P(A ∩ B1) + P(A ∩ B2) + ... + P(A ∩ Bn).
- Applying the definition of conditional probability, P(A ∩ Bi) = P(A | Bi) P(Bi).
- Substituting this into the sum yields the final result: P(A) = Σ P(A | Bi) P(Bi).
Define the terms outcome, event, sample space, mutually exclusive events. Consider the experiment of rolling of two fair dice simultaneously and represent its sample space. Also give examples of terms mentioned above related to this experiment.
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Consider the experiment of tossing two dice simultaneously. If X denotes the sum of two faces, find the probability for X ≤ 6.
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Derive the Poisson density function and find mean and variance.
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Find the Moment generating function of a uniform random variable distributed over (A, B) and find its first and second moments about origin, from the Moment generating function.
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Obtain the moment generating function of a uniformly distributed random variable.
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State and prove the density function of sum of two random variables.
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Let Z=X+Y-C, where X and Y are independent random variables with variance σx2, σy2 and C is constant. Find the variance of Z in terms of σx2, σy2 and C.
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Two statistically independent random variables X and Y have mean values E[X] = 2 and E[Y] = 4. They have second moments E[X2] = 8 and E[Y2] = 25. Find Variance of W = 3X-Y.
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If X is a random variable with mean 3 and variance 2, verify that the random Variables ‘X’ and Y= -6X+22 are orthogonal.
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State and prove the auto correlation and cross correlation function properties.
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A Gaussian RP has an auto correlation function RXX(τ) = (6 sin(πτ)) / (πτ). Determine a covariance matrix for the Random variable X(t).
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Derive the relation between PSDs of input and output random process of an LTI system.
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