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Chapter 13: Problem 25

At the 18 th hole of a golf course the probability that a golfer will score a par four is \(0.55\), the probability of one under is \(0.17\), of two under is \(0.03\), of one over is \(0.2\) and of two over is \(0.05\). Plot the (cumulative) distribution function.

Short Answer

Expert verified
Plot the step function of cumulative probabilities for scores -2, -1, 0, 1, 2.

Step by step solution

01

Understand the Data

We are given the probability of different scores for a golfer at the 18th hole. These scores can be considered as random variables with their probabilities: - Par four (\( X = 0 \)) has a probability of \( P(X=0) = 0.55 \).- One under (\( X = -1 \)) has a probability of \( P(X=-1) = 0.17 \).- Two under (\( X = -2 \)) has a probability of \( P(X=-2) = 0.03 \).- One over (\( X = 1 \)) has a probability of \( P(X=1) = 0.2 \).- Two over (\( X = 2 \)) has a probability of \( P(X=2) = 0.05 \).
02

Calculate Cumulative Probabilities

Calculate the cumulative distribution function (CDF) for each score. This means accumulating the probabilities from the lowest score to the highest:- For \( X = -2 \), \( F(-2) = P(X \leq -2) = 0.03 \).- For \( X = -1 \), \( F(-1) = P(X \leq -1) = 0.03 + 0.17 = 0.20 \).- For \( X = 0 \), \( F(0) = P(X \leq 0) = 0.20 + 0.55 = 0.75 \).- For \( X = 1 \), \( F(1) = P(X \leq 1) = 0.75 + 0.2 = 0.95 \).- For \( X = 2 \), \( F(2) = P(X \leq 2) = 0.95 + 0.05 = 1.00 \).
03

Plot the CDF Graph

Use the cumulative probabilities to plot the CDF. On the x-axis, plot the scores \(-2, -1, 0, 1, 2\). On the y-axis, plot the corresponding cumulative probabilities \(0.03, 0.20, 0.75, 0.95, 1.00\). Connect these points with a step function that rises with each point according to the cumulative probability.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution
In probability theory, a probability distribution describes how the values of a random variable are distributed. It gives us the probabilities of outcomes for an experiment or event. For instance, in golfing, different scores can occur, such as scoring a par, one under, or one over, each having a specific probability.
  • A par score, which is the expected score for a hole, is considered a baseline.
  • One under means the score is one less than par, often referred to as a birdie.
  • One over means the player took an extra hit, scoring a bogey.
Each outcome has an assigned probability, indicating the likelihood of its occurrence. Understanding probability distribution is key to determining which scores are most likely and how they influence the overall game's dynamics.
Random Variables
Random variables are a foundational concept in probability and statistics. They are basically functions that assign numerical values to each event in a sample space. For the golf score example, each possible score (like par, under, or over) is a random variable.
  • These scores are denoted by symbols, such as \( X = 0 \) for par or \( X = -1 \) for one under par.
  • The probability of each event is the likelihood that the random variable takes on that value.
Random variables help us quantify uncertain events, making it easier to calculate probabilities and understand the behavior of different outcomes within a given context.
Cumulative Probability
Cumulative probability refers to the probability that a random variable is less than or equal to a specific value. It's a crucial concept in evaluating the likelihood of a range of outcomes.
  • The cumulative distribution function (CDF) sums up the probabilities of all outcomes less than or equal to a critical value.
  • In the golf example, calculating cumulative probability helps determine the probability of achieving a score of par or better.
To construct a CDF, you begin with the lowest score and add up the probabilities sequentially until all are included. This method provides a complete view of how likely one is to attain any score equal to or below a given number.
Plotting Graphs
Plotting graphs, particularly cumulative distribution functions, is an invaluable tool for visualizing data distributions. Here鈥檚 how it works using our golf score example:
  • The x-axis represents different possible scores, and the y-axis represents their cumulative probabilities.
  • You plot points for cumulative probabilities associated with each score. For example, \( (-2, 0.03), (-1, 0.20), \) and so on.
  • The points are then connected using a step-like line, meaning it climbs to the next level as you progress through scores.
Visualizing data this way makes patterns and trends more apparent, allowing one to make informed decisions or predictions based on the probabilities of different outcomes.

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Most popular questions from this chapter

The wave amplitude \(X\) on the sea surface often has the following (Rayleigh) distribution: $$ f_{X}(x)=\left\\{\begin{array}{cl} \frac{x}{a} \exp \left(\frac{-x^{2}}{2 a}\right) & (x>0) \\ 0 & \text { (otherwise) } \end{array}\right. $$ where \(a\) is a positive constant. Find the distribution function and hence the probability that a wave amplitude will exceed \(5.5 \mathrm{~m}\) when \(a=6\).

Suppose that a coin is tossed three times and that the random variable \(W\) represents the number of heads minus the number of tails. (a) List the elements of the sample space \(S\) for the three tosses of the coin, and to each sample point assign a value \(w\) of \(W\). (b) Find the probability distribution of \(W\), assuming that the coin is fair. (c) Find the probability distribution of \(W\), assuming that the coin is biased so that a head is twice as likely to occur as a tail.

If a card is drawn from a well-shuffled pack of fifty-two playing cards, what is the probability of drawing (a) a red king (b) a \(3,4,5\) or 6 (c) a black card (d) a red ace or a black queen?

Find the distribution of the sum of the numbers when a pair of dice is tossed.

A Geiger counter and a source of radioactive particles are so situated that the probability that a particle emanating from the radioactive source will be registered by the counter is \(1 / 10000\). Assume that during the time of observation, 30000 particles emanated from the source. What is the probability that the number of particles registered was (a) zero, (b) three, (c) more than five?
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