R059210401 Probability Theory And Stochastic Process Feb08

Set No. 1

Code No: R059210401

II B.Tech I Semester Supplimentary Examinations, February 2008 PROBABILITY THEORY AND STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering, Electronics & Telematics and Electronics & Computer Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Define probability with an Axiomatic Approach. (b) In a box there are 100 resistos having resistance and tolerance as shown in table 1. If a resistor is chosen with same likelihood of being chosen for the three events, A as “draw a 470 Ω resistor”, B as “draw a 100 Ω resistor”, determine joint probabilities and conditional probabilities. Table 1 Number of resistors in a box having given resistance and tolerance. Resistance (Ω) 22 47 100 Total

[6+10]

Tolerance 5% 10% Total 10 14 24 28 16 44 24 8 32 62 38 100

2. (a) Define and explain the following density functions i. ii. iii. iv.

Binomial Exponential Uniform Rayleigh.

(b) What is density function of a random variable x, if x is guassian.

[12+4]

3. (a) A random variable X has a characteristic function given by Φx (ω) =



1 − |ω| |ω| ≤ 1 . Find density function 0 |ω| > 1

(b) A random variable X has the density function fX (x) = Find E[X], E[X 2 ] and variance.

1 −b|x| e a

−∞ ≤ x ≤ ∞ . [8+8]

4. The joint space for two random variables X and Y and corresponding probabilities are shown in table X,Y 1,1 2,2 3,3 4,4 P 0.2 0.3 0.35 0.15 Find and Plot 1 of 2

Set No. 1

Code No: R059210401 (a) FXY (x, y) , (b) marginal distribution functions of X and Y, (c) Find P(X ≤ 2, Y ≤ 2) and (d) Find P(1 < X ≤ 3, Y ≥ 3).

[5+5+3+3]

5. (a) let Y = X1 + X2 + ............+XN be the sum of N statistically independent random variables Xi , i=1,2.............. N. If Xi are identically distributed then find density of Y, fy (y). (b) Consider random variables Y1 and Y2 related to arbitrary random variables X and Y by the coordinate rotation. Y1 =X Cos θ + Y Sin θ Y2 = -X Sin θ + Y Cos θ i. Find the covariance of Y1 and Y2 , CY1Y2 ii. For what value of θ, the random variables Y1 and Y2 uncorrelated. [8+8] 6. (a) Consider random processes X(t) = A cos (ω0 t +θ) Y(t) = B cos (ω1 t +φ) Where A,B, ω1 and ω0 are constants, while θ and φ are statistically independent random variables each uniform on (0,2Π) i. Show that X(t) and Y(t) are jointly wide sense stationary ii. If θ = φ show that X(t) and Y(t) are not jointly wide sense stationary unless ω1 = ω0 . [8+8] (b) Let X(t) be the sum of a deterministic signal S(t) and a wide sense stationary noise process N(t). Find the mean value, auto correlation and auto covariance functions of X(t). Discuss the stationary of X(t). 7. (a) The PSD of random process is given by  π, |ω| <| δXX (ω) = 0, elsewhere Find its Auto correlation function. (b) State and Prove any four properties of PSD.

[8+8]

3 8. A random noise X(t) having power spectrum SXX (ω) = 49+ω 2 is applied to a to a network for which h(t) = u(t)t2 exp(−7t). The network response is denoted by Y(t)

(a) What is the average power is X(t) (b) Find the power spectrum of Y(t) (c) Find average power of Y(t).

[5+6+5] ⋆⋆⋆⋆⋆

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Set No. 2

Code No: R059210401

II B.Tech I Semester Supplimentary Examinations, February 2008 PROBABILITY THEORY AND STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering, Electronics & Telematics and Electronics & Computer Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Define and Explain the following with example: i. Sample Space ii. Discrete Sample Space iii. Continuous Sample Space (b) A pack contains 4 white and 2 green pencils, another contains 3 white and 5 green pencils. If one pencil is drawn from each pack, find the probability that i. both are white and ii. one is white and another is green.

[12+4]

2. (a) What are the conditions for the function to be a random variable? Discuss (b) What do you mean by continuous and discrete random variable? (c) Demonstrate with an examle that the probability of occurrence of any discrete value of a continuous random variable is zero. [4+6+6] 3. (a) Define moment generating function. (b) State properties of moment generating function. (c) Find the moment generating function about origin of the Poisson distribution. [3+4+9] 4. (a) State and prove central limit theorem. (b) Find the density of W = X + Y where the densities of X and Y are assumed to be fX (x) == [u(x) − u(x − 1)], fY (y) = [u(y) − u(y − 1)], [8+8] 5. (a) let Xi , i = 1,2,3,4 be four zero mean Gaussian random variables. Use the joint characteristic function to show that E {X1 X2 X3 X4 } = E[X1 X2 ] E[X3 X4 ] + E[X1 X3 ]E[X2 X4 ] + E[X2 X3 ] E[X1 X4 ] (b) Show that two random variables X1 and X2 with joint pdf. fX1X2 (X1 , X2 ) = 1/16 |X1 | < 4, 2 < X2 < 4 are independent and orthogonal.[8+8] 6. (a) State the properties of cross correlation function (b) Let X(t) be a wide sense stationary random process with auto correlation function. RXX (τ ) = e−a|τ | where a > 0 is a constant. Assume X(t) “amplitude modulates” a “carrier” Cos (ωo t + θ) as shown in figure 6, where ω0 is a 1 of 2

Set No. 2

Code No: R059210401

constant and θ is a random variable uniform on (-Π, Π) that is statistically independent of X(t). Determine the auto correlation function of Y(t). [6+10]

Figure 6 7. (a) The PSD of random process is given by  π, |ω| <| δXX (ω) = 0, elsewhere Find its Auto correlation function. (b) State and Prove any four properties of PSD.

[8+8]

3 8. A random noise X(t) having power spectrum SXX (ω) = 49+ω 2 is applied to a to a 2 network for which h(t) = u(t)t exp(−7t). The network response is denoted by Y(t)

(a) What is the average power is X(t) (b) Find the power spectrum of Y(t) (c) Find average power of Y(t).

[5+6+5] ⋆⋆⋆⋆⋆

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Set No. 3

Code No: R059210401

II B.Tech I Semester Supplimentary Examinations, February 2008 PROBABILITY THEORY AND STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering, Electronics & Telematics and Electronics & Computer Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Define an event and when do we say the events are mutually exclusive. Explain with an example. (b) When two dice are thrown determine the probabilities from axiom 3 for the following three events. i. ii. iii. iv. v.

A = {sum = 7} B = {8 ≺ sum ≤ 11} C = {10 ≺ sum} and determine P(B ∩ C) P(B ∩ C)

[8+8]

2. (a) In an experiment of fair wheel of chance is spun with the numbers 0 to 12 on the wheel. What is the distribution and density function explain with plot. (b) What is the distribution function of mixed random variable? Discuss what do you mean by density function. [8+8] 3. (a) Find the nth moment of uniform random variable and hence its mean. (b) Find the density function a random variable X whose characteristic function is ΦX (ω) = 12 e−|ω| −∞ ≤ ω ≤ ∞ . [8+8] 4. The joint space for two random variables X and Y and corresponding probabilities are shown in table Find and Plot (a) FXY (x, y) (b) marginal distribution functions of X and Y. (c) Find P(0.5 < X < 1.5), (d) Find P(X ≤ 1, Y ≤ 2) and (e) Find P(1 < X ≤ 2, Y ≤ 3). X, Y P

1,1 2,2 3,3 4,4 0.05 0.35 0.45 0.15

[3+4+3+3+3]

5. (a) For two zero mean Gaussian random variables X and Y show that their joint characteristic function is φXY(ω1 ,ω2 ) = exp{−1/2[σX2 ω12 +2ρσX σY ω1 ω2 +σY2 ω22 ]}. 1 of 2

Set No. 3

Code No: R059210401

(b) Statistically independent random variables X and Y have moments m10 = 2, m20 = 14, m02 = 12 and m11 = -6 find the moment µ 22. (c) Two Gaussian random variables X and Y have variances σX2 = 9 and σY2 = 4, respectively and correlation coefficient ρ. It is known that a coordinate rotation by an angle Π/8 results in new random variables Y1 and Y2 that are uncorrelated. what is ρ. [8+4+4] 6. A Random process is defined by X(t)= X0 +Vt where X0 and V are statistically independent random variables uniformly distributed on intervals . [X01 ,X02 ] and [V1 ,V2 ]. Respectively. Find (a) mean (b) the auto correlation (c) the auto covariance functions of X(t). (d) is X(t) stationary in any sense if so state the type. 7. (a) If the PSD of X(t) is Sxx(ω). Find the PSD of

[16]

dx(t) dt

(b) Prove that Sxx (ω) = Sxx (-ω ) (c) If R (τ ) = ae−b |τ |. Find the spectral density function, where a and b are constants. [5+5+6] 8. (a) For the network shown in figure 8 find the transfer function of the system.[8] (b) Define the following systems i. ii. iii. iv.

LTI system Causal system stable system Noise Bandwidth.

[4×2=8]

Figure 8 ⋆⋆⋆⋆⋆

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Set No. 4

Code No: R059210401

II B.Tech I Semester Supplimentary Examinations, February 2008 PROBABILITY THEORY AND STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering, Electronics & Telematics and Electronics & Computer Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) show that two elements cannot be both mutually exclusive and statistically independent. What is the condition for two events to be independent? (b) One card is selected from an ordinary 52-card deck with an event A as“select a king”, B as “select a jack or queen” and c as “select a heart”. Determine whether A,B and C are independent by pairs. (c) What is the probability of drawing 3 white and 4 green balls from a bag that contains 5 white and 6 green balls, if 7 balls are drawn simultaneously at random? [6+6+4] 2. (a) In an experiment of fair wheel of chance is spun with the numbers 0 to 12 on the wheel. What is the distribution and density function explain with plot. (b) What is the distribution function of mixed random variable? Discuss what do you mean by density function. [8+8] 3. (a) State and prove properties of moment generating function of a random variable X (b) The characteristic function for a random variable X is given by ΦX (ω) = 1 . Find mean and second moment of X. [8+8] (1−j2ω)N/2 4. The joint space for two random variables X and Y and corresponding probabilities are shown in table Find and Plot (a) FXY (x, y) (b) marginal distribution functions of X and Y. (c) Find P(0.5 < X < 1.5), (d) Find P(X ≤ 1, Y ≤ 2) and (e) Find P(1 < X ≤ 2, Y ≤ 3). X, Y P

1,1 2,2 3,3 4,4 0.05 0.35 0.45 0.15

[3+4+3+3+3]

¯ = 1andY ¯ = 2, variances σx2 = 4 and σY2 = 1 5. Two random variables X and Y have means X and a correlation coefficient ρXY = 0.4 New random variables W and V are defined by V=-X+2Y & W = X+3Y. Find 1 of 2

Set No. 4

Code No: R059210401 (a) The means (b) The variances (c) The correlations and (d) The correlation coefficient ρvw of V and W.

[16]

6. Let X(t) be a stationary continuous random process that is differentiable. Denote ˙ its time derivative by X(t). h• i (a) Show that E × (t) = 0. (b) Find R××˙ (τ ) in terms of R×× (τ )sss 7. (a) If the PSD of X(t) is Sxx(ω). Find the PSD of

[8+8] dx(t) dt

(b) Prove that Sxx (ω) = Sxx (-ω ) (c) If R (τ ) = ae−b |τ |. Find the spectral density function, where a and b are constants. [5+5+6] 3 8. A random noise X(t) having power spectrum SXX (ω) = 49+ω 2 is applied to a to a 2 network for which h(t) = u(t)t exp(−7t). The network response is denoted by Y(t)

(a) What is the average power is X(t) (b) Find the power spectrum of Y(t) (c) Find average power of Y(t).

[5+6+5] ⋆⋆⋆⋆⋆

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