Problem 7
Question
Suppose \(X\) is a random variable with mean 10 and variance \(9 .\) What can you say about \(P(|X-10| \geq 5) ?\)
Step-by-Step Solution
Verified Answer
\(P(|X - 10| \geq 5) \leq 0.36\).
1Step 1: Identify the Required Probability
We need to find the probability that the absolute deviation of the random variable \(X\) from its mean is at least 5, which is mathematically represented as \(P(|X - 10| \geq 5)\).
2Step 2: Apply Chebyshev's Inequality
Chebyshev's inequality states that for any random variable \(X\) with mean \(\mu\) and variance \(\sigma^2\), the probability that \(X\) deviates from \(\mu\) by more than \(k\sigma\) is at most \(\frac{1}{k^2}\). Here, \(\mu = 10\), \(\sigma = 3\) (since variance \(= 9\)), and we want \(P(|X - 10| \geq 5)\).
3Step 3: Calculate k Value
Determine \(k\) by calculating \(k\sigma\) such that \(k\sigma = 5\). With \(\sigma = 3\), we solve for \(k\) as follows: \(k \times 3 = 5\), yielding \(k = \frac{5}{3}\).
4Step 4: Evaluate the Probability Bound
Using Chebyshev’s inequality, \(P(|X - 10| \geq 5) \leq \frac{1}{k^2}\). Substitute \(k = \frac{5}{3}\) to get: \(P(|X - 10| \geq 5) \leq \left(\frac{3}{5}\right)^2 = \frac{9}{25}\).
5Step 5: Conclude the Result
Thus, by Chebyshev's inequality, \(P(|X - 10| \geq 5) \leq 0.36\).
Key Concepts
Random VariableProbabilityVariance
Random Variable
In the world of statistics and probability, a random variable is a fundamental concept. It's essentially a variable whose possible values are outcomes of a random phenomenon. Imagine rolling a fair six-sided die. The random variable could represent the number you get, which could be any integer from 1 to 6.
Random variables are classified into two types: discrete and continuous. A discrete random variable, like our die-rolling example, has a finite or countable number of possible outcomes. On the other hand, a continuous random variable has an infinite number of possible values, typically representing quantities that can change smoothly, like height or temperature.
In the context of Chebyshev's Inequality, we deal with a random variable that has an expected mean and variance. For instance, in our exercise, the random variable X has been assigned a mean of 10. This means if you were to observe or simulate this random variable many times, you'd expect the average outcome to be around 10.
Random variables are classified into two types: discrete and continuous. A discrete random variable, like our die-rolling example, has a finite or countable number of possible outcomes. On the other hand, a continuous random variable has an infinite number of possible values, typically representing quantities that can change smoothly, like height or temperature.
In the context of Chebyshev's Inequality, we deal with a random variable that has an expected mean and variance. For instance, in our exercise, the random variable X has been assigned a mean of 10. This means if you were to observe or simulate this random variable many times, you'd expect the average outcome to be around 10.
Probability
Probability is the measure of the likelihood that an event will occur. It ranges between 0 and 1, where 0 indicates an impossibility, and 1 indicates certainty. When we talk about probabilities related to random variables, we're essentially looking at how likely certain outcomes or range of outcomes are.
In our exercise, the focus is on finding the probability that the random variable differs from its mean (10) by at least 5. This is captured as the probability statement: \( P(|X - 10| \geq 5) \).
To estimate this probability efficiently, we apply Chebyshev's Inequality, which provides a way to find an upper limit on the probability that the random variable deviates a certain amount from its mean. This inequality is incredibly useful when we don't know much about the distribution of the random variable beyond its mean and variance.
In our exercise, the focus is on finding the probability that the random variable differs from its mean (10) by at least 5. This is captured as the probability statement: \( P(|X - 10| \geq 5) \).
To estimate this probability efficiently, we apply Chebyshev's Inequality, which provides a way to find an upper limit on the probability that the random variable deviates a certain amount from its mean. This inequality is incredibly useful when we don't know much about the distribution of the random variable beyond its mean and variance.
Variance
Variance is a statistical measurement that describes how much the values of a random variable differ from the mean. Think of it as a measure of the spread or dispersion within a set of data. A larger variance indicates that the numbers are quite spread out from the mean, while a smaller variance signifies that they are closer to the mean.
Mathematically, variance is denoted as \( \sigma^2 \) and is calculated as the average of the squared differences from the mean. In the context of our exercise, the variance of the random variable \( X \) is given as 9. This implies that, on average, the squared distance of \( X \)'s values from its mean (10) is 9.
Variance plays a crucial role in applying Chebyshev's Inequality. Because Chebyshev's Inequality needs the standard deviation (the square root of the variance) to calculate deviations, understanding how variance works helps us grasp why and how we use this inequality to find probabilities about how data is distributed around the mean.
Mathematically, variance is denoted as \( \sigma^2 \) and is calculated as the average of the squared differences from the mean. In the context of our exercise, the variance of the random variable \( X \) is given as 9. This implies that, on average, the squared distance of \( X \)'s values from its mean (10) is 9.
Variance plays a crucial role in applying Chebyshev's Inequality. Because Chebyshev's Inequality needs the standard deviation (the square root of the variance) to calculate deviations, understanding how variance works helps us grasp why and how we use this inequality to find probabilities about how data is distributed around the mean.
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