A filling station is supplied with gasoline once a week. If...

A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function $$f(x)=\left\{\begin{array}{ll}5(1-x)^{4} & 0<x<1 \\0 & \text { otherwise }\end{array}\right.$$ what must the capacity of the tank be so that the probability of the supply's being exhausted in a given week is .01?

A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function
$$f(x)=\left\{\begin{array}{ll}5(1-x)^{4} & 0<x<1 \\0 & \text { otherwise }\end{array}\right.$$ what must the capacity of the tank be so that the probability of the supply's being exhausted in a given week is .01?

Key Concepts

Quantiles and Inverse CDF

Quantiles, or inverse CDFs, are values below which a specified percentage of the data falls. Determining the required quantile involves setting the cdf equal to a given probability, such as 0.99, and solving for the corresponding value. This is important in risk assessment and capacity planning, where a threshold is set based on a desired probability.

Tail Probability

Tail probability refers to the probability that a random variable falls beyond a certain threshold, typically in the extreme ends ('tails') of the probability distribution. In many practical applications, such as capacity planning, ensuring that the tail probability (the chance of exceeding or falling below a specific value) is within acceptable limits is critical.

Probability Density Function

A probability density function (pdf) is a function that describes the relative likelihood for a continuous random variable to take on a given value. It is fundamental for defining the distribution of the variable and allows for the calculation of probabilities by integrating the function over an interval.

Cumulative Distribution Function

The cumulative distribution function (cdf) of a continuous random variable is obtained by integrating its pdf up to a given point. The cdf represents the probability that the random variable will take on a value less than or equal to that point. It is key in determining probabilities for intervals and in solving for specific percentiles or quantiles.

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