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

step 1 Here on this problem, we were wanting to find a few different probabilities using shibby -ships.

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

A random variable $X$ has a mean $\mu=10$ and a variance $\sigma^{2}=4$. Using Chebyshev's theorem, find
(a) $P(|X-10| \geq 4)$
(b) $P(|X-10|<4)$
(c) $P(4<X<16)$
(d) the value of the constant $c$ such that $P(|X-10| \geq$ $c) \leq 0.05$

Key Concepts

Interval Interpretation of Absolute Deviation

Interpreting absolute deviation conditions, such as |X - ?| ? k? or |X - ?| < k?, involves defining intervals around the mean within which the random variable's values are likely to fall. These interval interpretations facilitate the conversion of absolute deviation bounds into probability statements, which can then be estimated or bounded using results like Chebyshev's inequality.

Variance and Standard Deviation

Variance is the measure of how much the values of a random variable spread out from the mean, calculated as the average squared deviation from the mean. The standard deviation, which is the square root of the variance, provides a scale for the typical deviation from the mean and is directly used in Chebyshev's inequality to quantify the relative spread.

Chebyshev's Inequality

Chebyshev's inequality is a fundamental result in probability theory that applies to any random variable with a finite mean and variance. It states that the probability that the value of a random variable deviates from its mean by more than k times its standard deviation is at most 1/k². This inequality provides a distribution-free bound, meaning it holds regardless of the underlying distribution’s shape.

Mean

The mean of a random variable is a measure of central tendency that represents the average value expected from the distribution. It is defined as the sum (or integral) of all possible values weighted by their probabilities, and it serves as the central point about which variability is measured.

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