Drunk Drivers The frequency of alcohol-related traffic...

Drunk Drivers The frequency of alcohol-related traffic fatalities has dropped in recent years but is still high among young people. Based on data from the National Highway Traffic Safety Administration, the age of a randomly selected, alcohol-impaired driver in a fatal car crash is a random variable with probability density function given by $f(t)=\frac{4.045}{t^{1.532}}$ for $t$ in $[16,80]$ . Find the following probabilities of the age of such a driver. Source: Traffic Safety Facts. (a) Less than or equal to 25 (b) Greater than or equal to 35 (c) Between 21 and 30 (d) Find the cumulative distribution function for this random variable. (e) Use the answer to part (d) to find the probability that a randomly selected alcohol-impaired driver in a fatal car crash is at most 21 years old.

Drunk Drivers The frequency of alcohol-related traffic fatalities has dropped in recent years but is still high among young people. Based on data from the National Highway Traffic Safety Administration, the age of a randomly selected, alcohol-impaired driver in a fatal car crash is a random variable with probability density function given by
$f(t)=\frac{4.045}{t^{1.532}}$ for $t$ in $[16,80]$ .
Find the following probabilities of the age of such a driver.
Source: Traffic Safety Facts.
(a) Less than or equal to 25
(b) Greater than or equal to 35
(c) Between 21 and 30
(d) Find the cumulative distribution function for this random variable.
(e) Use the answer to part (d) to find the probability that a randomly selected alcohol-impaired driver in a fatal car crash is at most 21 years old.

Key Concepts

Cumulative Distribution Function (CDF)

The cumulative distribution function provides the probability that a random variable will take a value less than or equal to a specified number. It is obtained by integrating the probability density function from the lower limit of the support up to that number, serving as a key tool for computing cumulative probabilities and understanding the distribution's behavior.

Random Variable

A random variable is a quantity whose outcome is subject to the uncertainty inherent in a random process. It allows us to model real-world processes by assigning numerical values to outcomes in a structured way, forming the basis for analyzing probability distributions and performing statistical calculations.

Probability Density Function (PDF)

A probability density function describes how the probability is distributed over the values of a continuous random variable. It is a non-negative function that integrates to one over the entire support of the variable, and the probability that the variable falls within any given interval is determined by the integral of the PDF over that interval.

Support of a Random Variable

The support of a random variable is the set of all possible values that the variable can assume with non-zero probability. In context, it defines the interval within which the integration of the probability density function is carried out to ensure that the total probability sums to one.

Integration to Compute Probabilities

Calculating probabilities for a continuous random variable involves integrating its probability density function over the interval of interest. This fundamental operation in probability theory allows us to determine the likelihood that the random variable falls within a specific range.

Transcript

00:01 All right, so here we have another probability distribution function, f of t, is 4 .045 over t to the 1 .53.

00:20 And this is good from t016 to 80.

00:25 Okay, so this is the probability distribution for the age of a person in alcohol -related.

00:34 Did fatal car crash.

00:37 And so the age isn't years, so age 16 to age 80 is what we're looking at.

00:43 And so part a says find the probability that an individual involved in such a car crash is under 25.

00:58 So we just look at this integral 16 to 25 of 4 .045.

01:07 Over t to the 1 .532 and so this is just going to be okay 4 .045 and then we need an antiderivative of 0 .5 through t to the 1 .5 through 2 so we have a negative we're going to add 1 .5 through 2 so we're going to divide by 0 .532 and then we have t to the negative 0 .532 evaluated from 16 to 25 and if we plug in those numbers we get about 0 .3676.

02:01 Okay and then b says the probability that an individual is greater than the age of 35 that's the integral from 35 to 80 all the way up to the max age of again what is this is 4 .045 over t to the 1 .532 this is just 4 .5 so negative we have the same antiderivative as before but this time we're evaluating from 35 to 80.

02:52 So this is, we'll plug in those values.

02:57 We'll get 0 .4081.

03:05 Okay, and then c says the probability that the age of the individual in the car crash is between 21 and 30.

03:18 So that's just the integral from 21 to 30 of this function...

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