probability,random variables

SLOVED PROBLEMS

2.1 Consider the experiment of throwing a fair die. Let X be the r.v. which assigns 1 if the number that appears is even and 0 if the number that appears is odd. (a) What is the range of X? (b) Find P(X = 1) and P(X = 0).

2.2 Consider the experiment of tossing a coin three times (Prob. 1.1). Let X be the r.v. giving the number of heads obtained. We assume that the tosses are independent and the probability of a head is p. (a) What is the range of X? (b) Find the probabilities P(X = 0), P(X = 1), P(X = 2), and P(X = 3). The sample space S on which X is defined consists of eight sample points (Prob. 1.1):

2.3 An information source generates symbols at random from a four-letter alphabet {a, b, c, d} with probabilities , and . A coding scheme encodes these symbols into binary codes as follows: Let X be the r.v. denoting the length of the code—that is, the number of binary symbols (bits). (a) What is the range of X? (b) Assuming that the generations of symbols are independent, find the probabilities P(X = 1), P(X = 2), P(X = 3), and P(X > 3). TEXT BOOK PG NO : 111

2.5. Verify Eq. (2.6). Let x1 < x2 . Then (X ≤ x1 ) is a subset of (X ≤ x2 ); that is, (X ≤ x1 ) ⊂ (X ≤ x2 ). Then, by Eq. (1.41), we have.

2.6 Verify (a) Eq. (2.10); (b) Eq. (2.11); (c) Eq. (2.12). (a) Since (X ≤ b) = (X ≤ a) ∪ (a < X ≤ b) and (X ≤ a) ∩ (a < X ≤ b) = ∅, we have.

2.7 Textbook pg no : 113

2.8. Let X be the r.v. defined in Prob. 2.3. (a) Sketch the cdf FX (x) of X and specify the type of X. (b) Find (i) P(X ≤ 1), (ii) P(1 < X ≤ 2), (iii) P(X > 1), and (iv) P(1 ≤ X ≤ 2). (a) From the result of Prob. 2.3 and Eq. (2.18), we have which is sketched in Fig. 2-15. The r.v. X is a discrete r.v. Textbook pg no 114

2.10 Consider the function given by (a) Sketch F(x) and show that F(x) has the properties of a cdf discussed in Sec. 2.3B. (b) If X is the r.v. whose cdf is given by F(x), find (i) , (ii) , (iii) P(X = 0), and (iv) . (c) Specify the type of X.

Textbook pg no :117

2.11 Textbook pg 118

2.12

2.13

2.15 Textbook 122

2.19 Textbook 125

2.20 Texbook 126

2.21 textbook no 126

2.22 Textbook pg no: 127

2.53 Textbook pg no : 151

2.54 Textbook pg no : 152

Practice Problems from Schaum Series (Functions of One Random Variable): Solved problems 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.8, 4.9, 4.14

problems 4.1 TextBook pg no : 245

problems 4.2 TextBook pg no : 246

problems 4.3 TextBook pg no : 247

problems 4.4 TextBook pg no : 249

problems 4.5 TextBook pg no : 250

problems 4.6 TextBook pg no : 251

problems 4.8 TextBook pg no : 252

problems 4.9 TextBook pg no : 253

problems 4.14 TextBook pg no : 254

Practice Problems from Schaum Series (Mean, Variance, Moments): Solved Problems: 2.27, 2.28, 2.32, 2.33, 2.35, 4.42, 4.43

Solved Problems: 2.27 pg no : 132

Solved Problems: 2.28 pg no : 134

Solved Problems: 2.32 pg no : 138

Solved Problems: 2.33 pg no : 138

Solved Problems: 2.35 pg no : 139

Solved Problems: 4.42 pg no : 280

Solved Problems: 4.43 pg no : 280