Problem 70 - An Introduction to Mathematical Statistics and Its Applications

Question

A teenager is trying to get a driver's license. Write out the formula for the pdf \(p_{x}(k)\), where the random variable \(X\) is the number of tries that he needs to pass the road test. Assume that his probability of passing the exam on any given attempt is \(0.10\). On the average, how many attempts is he likely to require before he gets his license?

Step-by-Step Solution

Answer

The pdf for the number of trials the teenager needs to pass the test is \(p_{x}(k) = (1 - 0.10)^{k - 1} * 0.10\). On average, he is expected to take 10 attempts to pass the test.

Step 1: Identify the type of distribution

Observing that the problem is related to the number of trials needed to obtain the first success, we can identify this as a case of geometric distribution.

Step 2: Write out the formula for the pdf of a Geometric Distribution

The probability mass function (pdf) for a geometric distribution is given by \(p_{x}(k) = (1 - p)^{k - 1}p\), where \(p\) is the probability of success in a single trial, and \(k\) is the number of trials.

Step 3: Substitute the given values into the formula

Substituting the given probability of passing, \(p = 0.10\), into the equation, we have \(p_{x}(k) = (1 - 0.10)^{k - 1} * 0.10\).

Step 4: Find the expected value or the average

The formula for the expected value (or the average value) for a geometric distribution is \(E[X] = 1 / p\). Substituting the value of \(p = 0.10\) into this formula gives the average number of trials expected, \(E[X] = 1 / 0.10 = 10\). Thus, the expected number of attempts the teenager is likely to require to pass the test is 10.

Key Concepts

Probability Mass Function

In probability theory, a probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. For a geometric distribution, the PMF describes the likelihood of needing a certain number of trials to achieve the first success. Consider our driving test example. The probability mass function for a geometric distribution is given by:

• \( p_{x}(k) = (1 - p)^{k - 1} \times p \)

where:

• \( p \) is the probability of success in a single trial (e.g., passing the test), and
• \( k \) is the number of trials we need to reach that first success.

For our teenager trying for a driver's license, where the probability \( p \) of passing on each trial is 0.10, the PMF becomes:

• \( p_{x}(k) = (1 - 0.10)^{k - 1} \times 0.10 \)

This formula helps determine the probability associated with each specific number of attempts needed to pass the exam.

Expected Value

Expected value, often referred to as the mean, is a measure used to predict the average outcome of a random process. It is found by multiplying each possible outcome by its probability and summing the results. In the context of a geometric distribution, the expected value tells us how many trials we can expect to conduct before achieving the first success.

• The formula for the expected value \( E[X] \) in a geometric distribution is:
  • \( E[X] = \frac{1}{p} \)
  where \( p \) is the probability of success on each trial.

For our exercise, substituting \( p = 0.10 \), the expected value becomes:

• \( E[X] = \frac{1}{0.10} = 10 \)

This indicates that, on average, the teenager will need about 10 attempts to pass the road test.

Random Variable

A random variable is a variable whose values depend on the outcomes of a random phenomenon. In statistical terms, a random variable can take on different numerical outcomes based on the random process it corresponds to. In our driving test scenario, the random variable \( X \) represents the number of attempts required by the teenager to pass the road test.

• Random variables can be:
  • Discrete, like in our example where \( X \) can take values 1, 2, 3, and so forth, representing integer trial counts.
  • Continuous, where possible outcomes could cover a range of values without specific steps.

In this exercise, since \( X \) indicates the count of attempts until the first success, it necessitates a discrete distribution model—such as the geometric distribution. The behavior of \( X \) is characterized by its probability distribution, which tells us about the likelihood of different possible numbers of attempts.