4.67 A random variable X has a mean p=10 and variance σ^2=4. Using Chebyshev's theorem, find (a)

step 1 All right, solution by Chibisat theorem, Bcura, k sigma, that's an over k square, same part A. G

4.67 A random variable X has a mean p=10 and variance σ^2=4....

4.67 A random variable $X$ has a mean $p=10$ and variance $\sigma^{2}=4$. Using Chebyshev's theorem, find
(a) $P(X-10 \mid \geq 3)$
(b) $\mathrm{P}(|\mathrm{X}-10|<3)$
(c) $P(b<X<15)$
(d) the value of the constant $\mathrm{c}$ such that
$$P(X-10 \mid \geq \mathrm{c}) \leq 0.04$$

Key Concepts

Standard Deviation

The standard deviation is the square root of the variance and provides a measure of spread in the same units as the random variable. It is a key concept when applying Chebyshev's theorem, as the theorem uses the number of standard deviations away from the mean to set bounds on probabilities.

Probability Bounds

Probability bounds refer to the limits on the probability that a random variable falls within or outside a specified range. Chebyshev's theorem is used to obtain such bounds without any assumptions about the underlying distribution. It allows for estimating the maximum probability that a random variable deviates significantly (in absolute terms) from its mean.

Chebyshev's Inequality

Chebyshev's Inequality is a fundamental result in probability theory that gives a bound on the probability that the value of a random variable deviates from its mean by more than a specified number of standard deviations. It applies to any random variable regardless of its distribution, and is particularly useful when the exact distribution is unknown. The inequality states that the probability of a deviation of at least k standard deviations is at most 1/k².

Mean and Variance

The mean of a random variable is a measure of its central tendency, representing the average outcome if an experiment were repeated many times. The variance, on the other hand, measures the dispersion of the random variable around the mean, quantifying the expected squared deviation from the mean. Together, these moments provide essential information about the behavior and spread of the distribution.

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