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Chapter 6: Problem 33

(a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section \(6.1 ;\) (c) compute the mean and standard deviation, using the methods of this section; and \((d)\) draw a graph of the probability distribution and comment on its shape. $$ n=10, p=0.5 $$

Short Answer

Expert verified
The mean is 5, and the standard deviation is 1.58. The probability distribution is symmetric around the mean.

Step by step solution

01

Constructing the Binomial Probability Distribution

List all possible values of the random variable and calculate their corresponding probabilities using the binomial probability formula \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \].Here, \[ n = 10 \] and \[ p = 0.5 \].First, calculate \[ \binom{n}{k} \] for \[ k = 0, 1, 2, \text{ ... }, 10 \]. Then, compute each probability.
02

Tabulating the Probabilities

List the probabilities \[ P(X = k) \] for each value of \[ k \] as follows:\[ P(X=0), P(X=1), P(X=2), \text{ ... }, P(X=10) \].We can tabulate it as follows:\[ \begin{array}{c|c} k & P(X = k) \ \hline 0 & 0.00098 \ 1 & 0.00977 \ 2 & 0.04395 \ 3 & 0.11719 \ 4 & 0.20508 \ 5 & 0.24609 \ 6 & 0.20508 \ 7 & 0.11719 \ 8 & 0.04395 \ 9 & 0.00977 \ 10 & 0.00098 \end{array} \]
03

Computing the Mean using Section 6.1 Method

The mean or expected value \[ \text{E}(X) \] of a binomial distribution is given by \[ \text{E}(X) = np \].Here, \[ n = 10 \] and \[ p = 0.5 \], so \[ \text{E}(X) = 10 \times 0.5 = 5 \].
04

Computing the Standard Deviation using Section 6.1 Method

The standard deviation \[ \text{SD}(X) \] of a binomial distribution is given by \[ \text{SD}(X) = \sqrt{np(1-p)} \].Here, \[ n = 10 \] and \[ p = 0.5 \], so \[ \text{SD}(X) = \sqrt{10 \times 0.5 \times (1-0.5)} = \sqrt{2.5} = 1.58 \].
05

Drawing a Graph of the Probability Distribution

Plot a histogram with \[ k \] values on the x-axis and \[ P(X = k) \] on the y-axis.Assess the shape of the distribution. Since \[ p = 0.5 \], the distribution should be symmetric around the mean \[ \text{E}(X) = 5 \].
06

Commenting on the Shape

Review the histogram. It should show a symmetric, bell-shaped curve centered at \[ k = 5 \]. This symmetry suggests a balanced probability of success and failure.

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Key Concepts

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

Expected Value Calculation
The expected value, also known as the mean, represents the 'average' outcome if the experiment were to be repeated a large number of times. For a binomial distribution, it is calculated using the formula \(E(X) = np\).
In this exercise, we have \(n = 10\), which is the number of trials, and \(p = 0.5\), which is the probability of success on each trial. Plugging these values into the formula, we get:
\[ E(X) = 10 \times 0.5 = 5 \].
This means that, on average, you can expect 5 successes in 10 trials when the probability of success in each trial is 0.5. This result is very intuitive because if you flip a fair coin 10 times, you would expect around 5 heads.
Standard Deviation Calculation
Standard deviation measures the spread of the distribution around the mean. For a binomial distribution, the standard deviation is calculated using the formula \(SD(X) = \sqrt{np(1-p)}\).
In this case, \(n = 10\) and \(p = 0.5\). Plugging these values into the formula, we get:
\[ SD(X) = \sqrt{10 \times 0.5 \times (1-0.5)} = \sqrt{2.5} \approx 1.58 \].
The standard deviation of approximately 1.58 tells us that most of the values of the distribution lie within 1.58 units of the mean (5) when the distribution is observed frequently.
This helps us understand the degree of spread or variability from the average number of successes.
Probability Distribution Graph
A probability distribution graph visually represents the likelihood of different outcomes in a binomial experiment.
In this exercise, you would plot the values of \(k\) (number of successes) on the x-axis and the corresponding probabilities, \(P(X = k)\), on the y-axis.
Since \(p = 0.5\), the probabilities are symmetric around the mean, making the distribution bell-shaped.
Here鈥檚 how you would observe the shape:
  • At \(k = 5\), the graph peaks because the probability of getting exactly 5 successes is higher.
  • As you move away from 5, the probabilities decrease symmetrically on either side.
  • The graph will look like a symmetric bell-shaped curve centered around \(k = 5\).
This symmetry indicates a balanced chance of achieving the different numbers of successes in 10 trials when the success probability is 0.5.

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Most popular questions from this chapter

State the criteria for a binomial probability experiment.

In the following probability distribution, the random variable \(X\) represents the number of marriages an individual aged 15 years or older has been involved in. $$ \begin{array}{ll} x & P(x) \\ \hline 0 & 0.272 \\ \hline 1 & 0.575 \\ \hline 2 & 0.121 \\ \hline 3 & 0.027 \\ \hline 4 & 0.004 \\ \hline 5 & 0.001 \end{array} $$ (a) Verify that this is a discrete probability distribution. (b) Draw a graph of the probability distribution. Describe the shape of the distribution. (c) Compute and interpret the mean of the random variable \(X\). (d) Compute the standard deviation of the random variable \(X\). (e) What is the probability that a randomly selected individual 15 years or older was involved in two marriages? (f) What is the probability that a randomly selected individual 15 years or older was involved in at least two marriages?

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