The random variable is the number of persons living in a randomly selected occupied housing unit. Its probability distribution is as fallow y 1 2 3 4 5 6 7 P ( Y = y ) 0.265 0.327 0.161 0.147 0.065 0.022 0.023 calculate the mean of the random variable Y calculate the standard deviation of the random variable Y
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
The random variable is the number of persons living in a randomly selected occupied housing unit. Its probability distribution is as fallow
y 1 2 3 4 5 6 7 P ( Y = y ) 0.265 0.327 0.161 0.147 0.065 0.022 0.023 calculate the mean of the random variable Y calculate the standard deviation of the random variable Y
![**Title: Understanding Probability Distribution of Random Variables**
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**Introduction to Random Variable Y**
The random variable \( Y \) represents the number of persons living in a randomly selected occupied housing unit. The probability distribution of \( Y \) is given as follows:
| \( y \) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-------------|-------|-------|-------|-------|-------|-------|-------|
| \( P(Y = y) \) | 0.265 | 0.327 | 0.161 | 0.147 | 0.065 | 0.022 | 0.013 |
**Tasks:**
*Please round your answers to three decimal places.*
**a) Calculate the mean (expected value) of the random variable \( Y \).**
The mean (expected value) of \( Y \) is denoted by \( E(Y) \) or \( \mu \).
**Formula:**
\[ E(Y) = \mu = \sum_{i} y_i \cdot P(Y = y_i) \]
Fill this expected value in the provided space.
**b) Calculate the standard deviation of the random variable \( Y \).**
The standard deviation of \( Y \) is denoted by \( \sigma(Y) \).
**Formula:**
\[ \sigma(Y) = \sqrt{\sum_{i} (y_i - \mu)^2 \cdot P(Y = y_i)} \]
Fill this standard deviation in the provided space.
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By understanding how to compute the mean and standard deviation of the random variable \( Y \), you can gain insights into the average number of persons per housing unit and the variability around this average. This information is useful for statistical analysis and decision-making in various fields such as urban planning and resource allocation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fffe96887-fbda-42a8-abec-8c2855a00602%2Fb41464db-28ed-44d3-b182-d77d3856b2e4%2Fog3j3v.jpeg&w=3840&q=75)
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