MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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### Probability Distributions and Calculations **Problem Statement:** 1. **Suppose** random variable \( X \sim \mathcal{N}(2,4) \) (normal distribution with mean 2 and variance 4). Suppose random variable \( Y \) follows the distribution below (0 and 1 are the only two possible outcomes for \( Y \)): \[ \Pr(Y = y) = \begin{cases} 0.4 & \text{if } y = 0 \\ 0.6 & \text{if } y = 1 \end{cases} \] What is \( \Pr(X=1) \) and \( \Pr(Y=1) \) respectively? **Explanation:** - **\( X \) Distribution:** \( X \) is normally distributed with a mean (\( \mu \)) of 2 and a variance (\( \sigma^2 \)) of 4. This distribution, denoted as \( \mathcal{N}(2,4) \), tells us about the spread and central tendency of \( X \). - **\( Y \) Distribution:** \( Y \) is a discrete random variable with two possible outcomes: 0 and 1. The probability distribution for \( Y \) is clearly specified: - \(\Pr(Y = 0) = 0.4\) - \(\Pr(Y = 1) = 0.6\) **Questions:** - **\( \Pr(X = 1) \):** The question asks for the probability that the normally distributed variable \( X \) equals 1. Since the normal distribution is continuous, \(\Pr(X = 1)\) is technically zero because continuous distributions measure the probability of intervals rather than exact values. - **\( \Pr(Y = 1) \):** According to the provided distribution, \(\Pr(Y = 1) = 0.6\). This problem examines understanding of both continuous and discrete probability distributions.
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Transcribed Image Text:### Probability Distributions and Calculations **Problem Statement:** 1. **Suppose** random variable \( X \sim \mathcal{N}(2,4) \) (normal distribution with mean 2 and variance 4). Suppose random variable \( Y \) follows the distribution below (0 and 1 are the only two possible outcomes for \( Y \)): \[ \Pr(Y = y) = \begin{cases} 0.4 & \text{if } y = 0 \\ 0.6 & \text{if } y = 1 \end{cases} \] What is \( \Pr(X=1) \) and \( \Pr(Y=1) \) respectively? **Explanation:** - **\( X \) Distribution:** \( X \) is normally distributed with a mean (\( \mu \)) of 2 and a variance (\( \sigma^2 \)) of 4. This distribution, denoted as \( \mathcal{N}(2,4) \), tells us about the spread and central tendency of \( X \). - **\( Y \) Distribution:** \( Y \) is a discrete random variable with two possible outcomes: 0 and 1. The probability distribution for \( Y \) is clearly specified: - \(\Pr(Y = 0) = 0.4\) - \(\Pr(Y = 1) = 0.6\) **Questions:** - **\( \Pr(X = 1) \):** The question asks for the probability that the normally distributed variable \( X \) equals 1. Since the normal distribution is continuous, \(\Pr(X = 1)\) is technically zero because continuous distributions measure the probability of intervals rather than exact values. - **\( \Pr(Y = 1) \):** According to the provided distribution, \(\Pr(Y = 1) = 0.6\). This problem examines understanding of both continuous and discrete probability distributions.
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