Suppose X is a random variable with finite mean and variance. For 50 ≤r<1, we define the r th pe

step 1 We have a normal distribution with mean 100 and standard deviation 15. We want to find the proba

Suppose X is a random variable with finite mean and...

Suppose $X$ is a random variable with finite mean and variance. For $50 \leq r<1$, we define the $r$ th percentile value $\pi_X(r)$ by $$ P\left[X \leq \pi_X(r)\right]=\frac{r}{100} . $$ Thus, the 90 th percentile value $\pi_X(90)$ is defined by $$ P\left[X \leq \pi_X(90)\right]=0.90 . $$ Show that $$ \pi_X(90) \leq E[X]+3 \sigma, $$ and $$ \pi_X(95) \leq E[X]+\sigma \sqrt{19} . $$

Suppose $X$ is a random variable with finite mean and variance. For $50 \leq r<1$, we define the $r$ th percentile value $\pi_X(r)$ by
$$
P\left[X \leq \pi_X(r)\right]=\frac{r}{100} .
$$
Thus, the 90 th percentile value $\pi_X(90)$ is defined by
$$
P\left[X \leq \pi_X(90)\right]=0.90 .
$$
Show that
$$
\pi_X(90) \leq E[X]+3 \sigma,
$$
and
$$
\pi_X(95) \leq E[X]+\sigma \sqrt{19} .
$$

Key Concepts

Chebyshev’s Inequality

Chebyshev’s Inequality is a fundamental result in probability that bounds the probability of deviations from the mean in terms of the variance. It guarantees that for any random variable with finite mean and variance, the probability of deviating more than a specified multiple of the standard deviation from the mean is limited, regardless of the underlying distribution. This inequality is powerful in deriving probabilistic bounds without detailed knowledge of the distribution.

Bounding Quantiles Using Chebyshev’s Inequality

This concept involves applying Chebyshev’s Inequality to obtain upper bounds on percentile values (or quantiles) of a random variable. By interpreting the tail probabilities in terms of multiples of the standard deviation from the mean, one can establish bounds on where certain percentiles lie. This technique is particularly useful when the precise distribution is unknown or when only moments (mean and variance) are available.

Percentiles

Percentiles indicate the values below which a certain percentage of the data falls in a probability distribution. They are defined through the cumulative distribution function, establishing thresholds that partition the probability mass. This concept is essential in statistics for summarizing distributions and analyzing extreme values or tail behaviors.

Expectation and Variance

The expectation (or mean) of a random variable is the long-run average outcome of a random process, while the variance measures the spread or dispersion of the data around the mean. The standard deviation, obtained by taking the square root of the variance, provides a scale for measuring deviations in the same units as the random variable.

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