Probability Random Variables Assignment

Probability Random Variables Assignment

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).

Probability Random Variables Assignment

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.

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Probability Random Variables Assignment

2.7 Textbook pg no : 113Graphical user interface, text Description automatically generated

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

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2.11 Textbook pg 118

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2.12

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2.13

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2.15 Textbook 122

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2.19 Textbook 125

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2.20 Texbook 126

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2.21 textbook no 126

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2.22 Textbook pg no: 127

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Probability Random Variables Assignment

2.53 Textbook pg no : 151

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2.54 Textbook pg no : 152

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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

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problems 4.2 TextBook pg no : 246

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problems 4.3 TextBook pg no : 247

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problems 4.4 TextBook pg no : 249

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problems 4.5 TextBook pg no : 250

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problems 4.6 TextBook pg no : 251

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problems 4.8 TextBook pg no : 252

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problems 4.9 TextBook pg no : 253

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problems 4.14 TextBook pg no : 254

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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

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Solved Problems: 2.28 pg no : 134

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Solved Problems: 2.32 pg no : 138

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Solved Problems: 2.33 pg no : 138

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Solved Problems: 2.35 pg no : 139

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Solved Problems: 4.42 pg no : 280

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Solved Problems: 4.43 pg no : 280

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