SUBJECT : PROBABILITY AND RANDOM PROCESSES CLASS : II B.E. - ECE
UNIT – I RANDOM VARIABLES PART A – QUESTIONS
1. If the random variable X takes the values 1,2,3 and 4 such that 2P(X = 1) = 3P(X = 2) = P(X = 3) = 5P(X = 4).
Find the probability distribution.
2. Define probability density function.
3. Write any two properties of cdf.
4.What do you mean by conditional probability?
5. If E(X) = 1 , E[X(X – 1)] = 4 find E(X2) ,V(X) and V(2 – 3X).
6. If F(x) = 1 – (1 + x)e - x, find f(x).
7. If f(x) = 1π 11+x2 , -¥ < x < ¥ ,find F(x).
8. Define probability mass function.
9. Define continuous random variable and give an example.
10.A continuous random variable X that can assume any value between x = 2 and x = 5 has density function given by
f(x) = k(1+x), find P(X < 4).
11.If X is a continuous random variable whose probability density function is given by f(x) = C(4x – 2x2) , 0 < x < 2
Find (a) the value of C (b) P(X > 1).
12.Find the MGF for the distribution where f(x) = 23 at x = 1
13 at x = 2
0 otherwise
13.If a random variable X has the MGF MXt=33-t , find the standard deviation of X.
14.The number of failures of a computer system in a week of operations has the following pmf:
No. of failures: 0 1 2 3 4 5 6
Probability : 0.18 0.28 0.25 0.18 0.06 0.04 0.01
15.The mean and variance of a binomial variate are 6 and 8 respectively. Find (i) P(X =1) (ii) P(X ≥ 2).
Find the mean number failures in a week.
16.If the Mgf of a uniform r.v X is 1t(e5t-e4t), t≠0 then find its mean.
17.If X is a Poisson variate such that E(x2) = 6, then find the value of E(x).
18.Define Poson distribution.
19.If X is a Poisson variate such that P(X = 2) = 9P(X = 4) + 90P(X = 6), find (i) mean of X (ii) variance of X.
20.The moment generating function of a random variable X is given by MXt=e3(et-1). Find P(X = 1).
21.Define Geometric distribution.
22.Let X be a continuous random variable with pdf f(x) = x2 in 1 < x < 5. Find the pdf of Y = 2X – 3 .
23. Let X be a continuous random variable with pdf f(x) = 2x, 0 < x < 1. Find the pdf of Y = 8X3 .
UNIT – I RANDOM VARIABLES PART B - QUESTIONS
1. A random variable X has the following probability function.
x / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7p(x) / 0 / k / 2k / 2k / 3k / k2 / 2k2 / 7k2 + k
(i) Find k (ii) Evaluate P(X < 6), P(X ≥ 6),and P(0 < X < 5) (iii) Determine the distribution function of X
(iv) Find the minimum value of λ for which P(X ≤λ ) > ½ .
2.The probability mass function of a random variable X is defined as P(X = 0) = 3C2 , P(X = 1) = 4C – 10C2 and P(X = 2) = 5C -1, where C > 0. Find (1) the value of C (2) the distribution of X (3) the cumulative distribution function of X.
3.The cdf of a rv X is given by
F(x) = 0 , x < 0
x2 , 0 ≤ x < ½
1-325(3-x)2 , ½ ≤ x < 3
1 , x ≥ 3
Find the pdf of X and evaluate P(⃓ x⃓ < 1).
4. If the density function of a continuous random variable X is given by
f(x) = ax , 0 ≤ x ≤ 1
a , 1 ≤ x ≤ 2
3a – ax , 2 ≤ x ≤ 3
0 , otherwise
(i) Find the value of ‘a’ (ii) Find the cdf of X (iii) P(X > 1.5).
5. The density function of a random variable X is given by f(x) = kx(2 – x), 0 ≤ x ≤ 2. Find K, mean, variance, and rth moment.
6. The density function of a random variable X is given by f(x) = 2(1 – x) , 1 < x < 2.Find mean, variance, and rth moment.
7. The probability function of an infinite discrete distribution is given by P(X = x) = 12x , x = 1,2,….∞. Find the mean and variance of the distribution. Also find P(X is even), P(X ≥ 5) and P(X is divisible by 5).
8. If f(x) = x , 0 ≤ x ≤ 1
2 – x , 1 ≤ x ≤ 2
0 , otherwise.
Find MGF, mean and variance.
9. Find the MGF , mean and variance of Binomial distribution.
10. Out of 800 families with 4 children each,how many families would be expected to have (i) 2 boys and 2 girls
(ii) At-least 1 boy (iii) At-most 2 girls (iv) Children of both sexes. Assume equal probabilities for boys and girls.
11.Six dice are thrown 729 times. How many times do you expect at-least 3 dice to show a five or a six.
12.Four coins were tossed simultaneously. What is the probability of getting (i) 2 heads (ii) atleast 2 heads
(iii) atmost 3 heads.
13.Derive the MGF mean and variance of Poisson distribution.
14.The number of monthly breakdown of a computer is a r.v having a Poisson distribution with mean equal to 1.8. Find the probability that this computer will function for a month with (i) without breakdown (ii) with only one
breakdown (iii) with atleast one breakdown.
15.A manufacturer of cotter pins knows that 5% of his product is defective. If he sells pins in boxes of 100 and guarantees that not more than 4 pins will be defective. What is the probability that a box will fail to meet the guaranteed quality?
16.Find the MGF , mean and variance of Geometric distribution.
17. If the probability that an applicant for a driver’s license will pass the road test on any given trials is 0.8,what is the probability that he will finally pass the test: (i) On the fourth trial (ii)In fewer than 4 trials.
18.A and B shoot independently until each has hit his own target. The probabilities of their hitting the target are 3/5 and 5/7 respectively. Find the probability that B will require more shots than A.
19.State and prove that memory less property of Geometric distribution.
20. A random variable X has a uniform distribution over (-3,3), compute (i) P(X<2), (ii) P(X<2),(iii) P(X-2<2)
(iv)Find K for which PXk=13.
21.Buses arrive at a specified bus stop at 15 min. intervals starting at 7.A.M., that is, they arrive at 7, 7:15, 7:30, 7:45 and so on. If a passenger arrives at the bus stop at a random time that is uniformly distributed between 7 and 7:30 A.M., find the probability that he waits (i) less than 5 min for a bus (ii) atleast 12 min for a bus.
22.Derive the MGF mean and variance of exponential distribution.
23.State and prove that memory less property of exponential distribution.
24.The daily consumption of milk in excess of 20,000 gallons in approximately exponentially distributed with θ = 3000. The city has a daily stock of 35,000 gallons. What is the probability that of two days selected at random, the stock is insufficient for both days.
25. The time (in hours) required to repair a machine is exponentially distributed with parameter λ=12.
(i) What is the probability that the repair time exceeds 2 hours?
(ii) What is the conditional probability that a repair takes 11h given that its duration exceeds 8h?
26.Obtain MGF, mean and Variance of Gamma distribution.
27.In a certain city the daily consumption of electric power in millions of kilowatt hours can be treated as a random variable
having an erlang distribution with parameters λ=12 and k = 3. If the power plant of this city has a daily capacity of 12 millions kilowatt hours, what is the probability that this power supply will be inadequate on any given day?
28.If X is normally distributed with mean 30 and standard deviation 5, find out the probability that (i) X ≥ 45
(ii) 26 ≤ X ≤ 40 (iii)X-30>5.
29. If X is exponentially distributed with parameter α, find the pdf of Y = log X .
30.If X is uniformly distributed over (-π2,π2). Find the pdf of Y = tan X .
UNIT – II TWO DIMENSIONAL RANDOM VARIABLES PART A – QUESTIONS
1. The joint pdf of R.V X and Y is given by f(x,y) = kxy e-(x2+y2) , x > 0, y > 0. Find the value of k.
2. Let X and Y have the joint pdf f(x,y) = 2, 0 < x < y< 1. Find the marginal and conditional density functions.
3. If f(x,y) = xy/96, 0 < x < 4, 1 < y < 5, find E(X).
4. Let X and Y be R.Vs having the joint pdf f(x,y) = 32 (x2+y2) , 0 < x < 1, 0 < y < 1, find E(xy)
5.If the joint pdf of (X,Y) is f(x,y) = ¼ , 0 ≤ x,y ≤ 2, find P(X+Y ≤ 1).
6. If f(x,y) = 2 , 0 < x < y < 1, find f(y/x).
7. X and Y are independent R.Vs with variance 2 and 3. Find the variance of 3X + 4Y.
8.The joint density functions of X and Y is f(x,y) = e-(x+y) , 0 ≤ x,y ≤ ¥ .Are X and Y independent? Also find P( X < 1).
9. If f(x,y) = x + y, 0 < x <1, 0 < y <1, check whether X and Y are independent.
10.If f(x,y)= ke-(2x+y) , x > 0, y > 0, find the value of ‘k’.
11.Write the acute angle between the two lines of regression.
12.Show that the correlation coefficient rxy takes values in the range-1 to 1.
13.Show that cov2(x,y) ≤ var(x). var(y).
14.The tangent of the angle between the lines of regression y on x and x on y is 0.6 and sx =sy/2 , find the correlation coefficient between x and y.
15.The correlation coefficient between two random variables X and Y is 0.6. If σX=1.5, σY=2,X=10 and Y=20. Find the regression line of Y on X.
16.State central limit theorem.
UNIT – II TWO DIMENSIONAL RANDOM VARIABLES PART B – QUESTIONS
1. The two dimensional R.V (X,Y) has the joint probability mass function f(x,y) = x+2y27 , x = 0,1,2 ; y = 0,1,2. Find the marginal and conditional distributions. Also find the conditional distribution of Y given X=1. Also find P(X + Y < 3).
2. If the joint pdf of X and Y is given by f(x,y) = k (6 – x - y), 0 < x <2, 2 < y < 4 , find (i)P(X < 1 ,Y < 3) (ii)P(X < 1 / Y < 3) (iii)P(X + Y < 3).
3.The joint pdf of a two dimensional R.V (X,Y) is given by f(x,y) = xy2 + x2/8 , 0 ≤ x ≤ 2, 0 ≤ y ≤ 1 . Compute (i) P(X >1/ Y < ½) (ii) P(Y < ½/ X >1) (iii) P(X < Y) (iv) P(X + Y ≤ 1).
4.The joint pdf of a two dimensional R.Vis f(x,y) =cx(x-y), 0 < x <2, -x < y < x . Find (i)fX(x) (ii) fY(y) and(iii) fYX(yx).
5. The joint pdf of (X,Y) is given by f(x,y) = kxye-(x2+y2), x > 0, y > 0. Find the value of k and also prove that X and Y are independent.
6. Suppose that the joint density function is given by f(x,y) = 65(x+ y2); 0 ≤ x ≤ 1; 0 ≤ y ≤ 1. Obtain the marginal pdf of X and that of Y. Hence find P14≤ y ≤ 34.
7.Two R.Vs X and Y have the following joint pdf f(x,y) = 2 - x – y , 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Find the correlation coefficient between X and Y.
8.Two R.Vs X and Y have the following joint pdf f(x,y) = x + y , 0 ≤ x ,y ≤ 1.Find the correlation coefficient between X and Y.
9.Find the correlation coefficient and regression equations between X and Y
X 55 56 58 59 60 60 62
Y 35 38 37 39 44 43 44
10. The two lines of regression are 8x – 10y + 66 = 0, 40x – 18 y – 214 = 0. Find (i) the mean values of X and Y (ii) correlation coefficient between X and Y.
11. Obtain the rank correlation for the following data:
X 68 64 75 50 64 80 75 40 55 64
Y 62 58 68 45 81 60 68 48 50 70
12.If the joint pdf of (X,Y) is given by fXY(x,y) = x + y , 0 ≤ x,y ≤ 1, find the pdf of U = XY.
13.If the joint density of X1 and X2 is given by f(x1,x2) = 6 e-3x1-2x2 for x1>0, x2>0. Find the density function of V = X1 + X2 .
14.If X and Y are independent random variables with pdf’s e-x, x≥0 and e-y,y≥0, respectively, find the density functions of U=XX+Y and V = X + Y. Are U and V independent?
15.If X1, X2,……..Xn are Poisson Variables with parameter l= 2, use central limit theorem to estimate P(120 < Sn < 160) where Sn = X1 + X2 + ……..+ Xn; and n = 75. Given 02.45fzdz = 0.4929 and 00.8fzdz = 0.2881.
16.The life time of a certain brand of a tube light may be considered as a random variable with mean 1200 h and standard deviation 250 h. Find the probability, using central limit theorem , that the average life time of 60 lights exceeds 1250 h.
UNIT – III RANDOM PROCESSES PART A – QUESTIONS
1. Define wide sense stationary process.
2. Define strict sense stationary process.
3. Define Markov process.
4. Define one step transition probability.
5. State Chapman-Kolmogrov theorem.
6. What are the postulates of Poisson process.
7. Prove that Poisson process is not stationary.
8. Prove that the difference of two independent Poisson process is not a Poisson process.
9. Define SSS process.
10.Classify Random processes.
11.What do you mean by evolutionary process.
12.State the postulates of Poisson process.
13.What do you mean by sine wave random process?
UNIT – III RANDOM PROCESSES PART B – QUESTIONS
1. At the receiver of an AM radio, the received signal containing cosine carrier signal at the carrier frequency w with a random phase q that is uniformly distributed over (0,2π). The receiver carrier signal is X(t) = A cos(wt + q), show that {X(t)} is a wide sense stationary process.
2. If A and B are two independent uncorrelated random variables with zero means and same variances then show that
X(t) = A cos wt + B sin wt is W.S.S.
3. The process {X(t) : t Î T} whose probability distribution , under certain conditions is given by