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(10) Find Std(X) when X is a discrete random variable with the following given density. (i) f(z) = , z = 6, 7, 8, 9, 10, (iii) f(x) = T,7 = 1,2, · . · , n, where n = 100, (9() (4) (vii) f(z) = 0.3(0.7)* , z = 0, 1, · .., (ix) f(x) = TD, z = 1, 2, - . . (v) f(x) = -, z = 0,1,2, 3,

(10) Find Std(X) when X is a discrete random variable with the following given density. (i) f(z) = , z = 6, 7, 8, 9, 10, (iii) f(x) = T,7 = 1,2, · . · , n, where n = 100, (9() (4) (vii) f(z) = 0.3(0.7)* , z = 0, 1, · .., (ix) f(x) = TD, z = 1, 2, - . . (v) f(x) = -, z = 0,1,2, 3,

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Answer last two sub parts of Q. 10.

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The image displays two multiple-choice questions, each with a set of possible answers. ### Question (vii) The options for this question are displayed in a horizontal line with corresponding radio buttons: - Option 1: \( \sqrt{7/3} \) - Option 2: \( 0.7/\sqrt{0.3} \) - Option 3: \( \sqrt{0.7}/0.3 \) - Option 4: \( 7/3 \) - Option 5: "None of the above" - Option 6: "N/A" (selected) ### Question (xi) The options for this question are similarly displayed in a horizontal line with radio buttons: - Option 1: "Does not exist" - Option 2: 1 - Option 3: 2 - Option 4: 3 - Option 5: 4 - Option 6: "N/A" (selected) Both questions have a yellow background, and the currently selected answers are indicated with a filled circle in the "N/A" options.
Transcribed Image Text:The image displays two multiple-choice questions, each with a set of possible answers. ### Question (vii) The options for this question are displayed in a horizontal line with corresponding radio buttons: - Option 1: \( \sqrt{7/3} \) - Option 2: \( 0.7/\sqrt{0.3} \) - Option 3: \( \sqrt{0.7}/0.3 \) - Option 4: \( 7/3 \) - Option 5: "None of the above" - Option 6: "N/A" (selected) ### Question (xi) The options for this question are similarly displayed in a horizontal line with radio buttons: - Option 1: "Does not exist" - Option 2: 1 - Option 3: 2 - Option 4: 3 - Option 5: 4 - Option 6: "N/A" (selected) Both questions have a yellow background, and the currently selected answers are indicated with a filled circle in the "N/A" options.
**Finding the Standard Deviation of a Discrete Random Variable** *Problem 10: Find \( \text{Std}(X) \) when \( X \) is a discrete random variable with the following given density functions:* (i) \( f(x) = \frac{1}{5}, \, x = 6, 7, 8, 9, 10. \) (ii) \( f(x) = \frac{1}{100}, \, x = 1, 2, \ldots, n, \) where \( n = 100. \) (iii) \( f(x) = \frac{1}{\binom{6}{3}} \binom{6}{x}, \, x = 0, 1, 2, 3. \) (iv) \( f(x) = 0.3 (0.7)^x, \, x = 0, 1, \ldots \) (v) \( f(x) = \frac{1}{x(x+1)}, \, x = 1, 2, \ldots \) **Explanation:** 1. **Probability Mass Functions (PMFs)**: - Each function \( f(x) \) represents a PMF of a discrete random variable \( X \). - The functions cover different probabilities for \( X \) taking certain integer values. 2. **Standard Deviation**: - The task is to calculate the standard deviation \( \text{Std}(X) \) for each random variable \( X \) using the given PMFs. - This involves finding the expected value \( E(X) \) and the variance \( \text{Var}(X) \), with \( \text{Std}(X) = \sqrt{\text{Var}(X)} \). 3. **Combinations and Factorials in (iii)**: - Explanation of combinations \( \binom{n}{k} \) is necessary, which denotes the number of ways to choose \( k \) items from \( n \) without regard to order. 4. **Geometric Distribution in (iv)**: - The function \( 0.3 (0.7)^x \) corresponds to the PMF of a geometric distribution that models the number of trials until the first success, here with a probability of success \( 0.3 \). 5. **
Transcribed Image Text:**Finding the Standard Deviation of a Discrete Random Variable** *Problem 10: Find \( \text{Std}(X) \) when \( X \) is a discrete random variable with the following given density functions:* (i) \( f(x) = \frac{1}{5}, \, x = 6, 7, 8, 9, 10. \) (ii) \( f(x) = \frac{1}{100}, \, x = 1, 2, \ldots, n, \) where \( n = 100. \) (iii) \( f(x) = \frac{1}{\binom{6}{3}} \binom{6}{x}, \, x = 0, 1, 2, 3. \) (iv) \( f(x) = 0.3 (0.7)^x, \, x = 0, 1, \ldots \) (v) \( f(x) = \frac{1}{x(x+1)}, \, x = 1, 2, \ldots \) **Explanation:** 1. **Probability Mass Functions (PMFs)**: - Each function \( f(x) \) represents a PMF of a discrete random variable \( X \). - The functions cover different probabilities for \( X \) taking certain integer values. 2. **Standard Deviation**: - The task is to calculate the standard deviation \( \text{Std}(X) \) for each random variable \( X \) using the given PMFs. - This involves finding the expected value \( E(X) \) and the variance \( \text{Var}(X) \), with \( \text{Std}(X) = \sqrt{\text{Var}(X)} \). 3. **Combinations and Factorials in (iii)**: - Explanation of combinations \( \binom{n}{k} \) is necessary, which denotes the number of ways to choose \( k \) items from \( n \) without regard to order. 4. **Geometric Distribution in (iv)**: - The function \( 0.3 (0.7)^x \) corresponds to the PMF of a geometric distribution that models the number of trials until the first success, here with a probability of success \( 0.3 \). 5. **
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Dear students the parts are different statement based question ,so they are treated as different question,so I have provided solution to only the first mentioned part.

 

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