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Chapter 6: Problem 7

A coin is tossed three times. Let \(X\) be the number of heads that turn up. Find \(V(X)\) and \(D(X)\)

Short Answer

Expert verified
\(V(X) = \frac{3}{4}\) and \(D(X) = \frac{\sqrt{3}}{2}\).

Step by step solution

01

Define the Random Variable

Let the random variable \(X\) represent the number of heads obtained when a coin is tossed three times.
02

Determine the Sample Space

The sample space for tossing a coin three times is \(\{HHH, HHT, HTH, THH, TTH, THT, HTT, TTT\}\).
03

Find the Probability Distribution of X

The possible values for \(X\) are 0, 1, 2, and 3. Calculate the probabilities: \[P(X=0) = \frac{1}{8},\ P(X=1) = \frac{3}{8},\ P(X=2) = \frac{3}{8},\ P(X=3) = \frac{1}{8}.\]
04

Calculate the Expected Value \(E(X)\)

Use the formula \(E(X) = \sum x_i P(X = x_i)\). Compute \[E(X) = 0 \cdot \frac{1}{8} + 1 \cdot \frac{3}{8} + 2 \cdot \frac{3}{8} + 3 \cdot \frac{1}{8} = \frac{3}{2}.\]
05

Compute the Variance \(V(X)\)

Variance is calculated using the formula \(V(X) = E(X^2) - [E(X)]^2\). First compute \[E(X^2) = 0^2 \cdot \frac{1}{8} + 1^2 \cdot \frac{3}{8} + 2^2 \cdot \frac{3}{8} + 3^2 \cdot \frac{1}{8} = \frac{15}{8}.\] Then find \[V(X) = \frac{15}{8} - \left(\frac{3}{2}\right)^2 = \frac{3}{4}.\]
06

Calculate the Standard Deviation \(D(X)\)

The standard deviation is the square root of the variance: \[D(X) = \sqrt{V(X)} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}.\]

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Key Concepts

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

Understanding Random Variables
In probability and statistics, a random variable is a variable that represents the outcomes of a random phenomenon. It is a way to map outcomes of a random process to numerical values.

In the context of our exercise, the random variable \(X\) is defined as the number of heads you get when you toss a coin three times. This numerical representation helps in analyzing the random event in a structured manner.
  • In this example, possible values for \(X\) are 0, 1, 2, and 3, which correspond to getting no heads, one head, two heads, and three heads, respectively.
  • Each outcome in the sample space like "HHH" or "THT" corresponds to one of these values of \(X\).
This lays the groundwork to calculate probabilities and further analyze the behavior of these outcomes.
Diving into Variance
Variance is a measure of how much the values of a random variable differ from the expected value, indicating the spread of the values.

In simpler terms, it's how much you expect the values to vary from the average. In our example, after calculating the expected value \(E(X)\), we compute the variance \(V(X)\) using the formula:
\[ V(X) = E(X^2) - [E(X)]^2 \]
  • First, calculate the expected value of squared outcomes, \(E(X^2)\).
  • Next, find the square of the expected value \([E(X)]^2\).
The resulting variance \( \frac{3}{4} \) provides insight into how much the count of heads obtained deviates from its mean value. A lower variance signifies outcomes are consistently near the expected value, while higher variance suggests more diversity in the outcomes.
Decoding Standard Deviation
Standard deviation is a commonly used measure of the amount of variation or dispersion of a set of values.

It is the square root of the variance and provides an average distance of each data point from the mean, in the same units as the data.
  • The formula for standard deviation is \(D(X) = \sqrt{V(X)}\).
  • In our coin toss example, \(D(X) = \frac{\sqrt{3}}{2}\), providing an easily interpretable measure of variability.
When you know the standard deviation, you can understand how much the number of heads is likely to vary from the expected number (mean). The smaller the standard deviation, the closer the data points tend to be to the mean.
Calculating Expected Value
The expected value is a fundamental concept in probability, often considered as the "average" outcome you can expect from a random process.

For a random variable \(X\), the expected value is computed by multiplying each possible value of \(X\) by its probability and summing up those products:
\[ E(X) = \sum x_i P(X = x_i) \]
  • In our exercise, this translates to using the probabilities and values derived from the possible outcomes of tossing a coin.
  • Example: Multiply each number of heads by its probability, then add them to find the expected value \(\frac{3}{2}\), or 1.5.
The expected value tells you what the "center" or "average" outcome in the long run would be. It serves as a vital baseline to understand where most of your outcomes will lie when you toss the coin multiple times.

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Most popular questions from this chapter

On September \(26,1980,\) the New York Times reported that a mysterious stranger strode into a Las Vegas casino, placed a single bet of 777,000 dollars on the "don't pass" line at the crap table, and walked away with more than 1.5 million dollars. In the "don't pass" bet, the bettor is essentially betting with the house. An exception occurs if the roller rolls a 12 on the first roll. In this case, the roller loses and the "don't pass" better just gets back the money bet instead of winning. Show that the "don't pass" bettor has a more favorable bet than the roller.

In a second version of roulette in Las Vegas, a player bets on red or black. Half of the numbers from 1 to 36 are red, and half are black. If a player bets a dollar on black, and if the ball stops on a black number, he gets his dollar back and another dollar. If the ball stops on a red number or on 0 or 00 he loses his dollar. Find the expected winnings for this bet.

An urn contains exactly 5000 balls, of which an unknown number \(X\) are white and the rest red, where \(X\) is a random variable with a probability distribution on the integers \(0,1,2, \ldots, 5000\) (a) Suppose we know that \(E(X)=\mu\). Show that this is enough to allow us to calculate the probability that a ball drawn at random from the urn will be white. What is this probability? (b) We draw a ball from the urn, examine its color, replace it, and then draw another. Under what conditions, if any, are the results of the two drawings independent; that is, does \(P(\) white, white \()=P(\text { white })^{2} ?\) (c) Suppose the variance of \(X\) is \(\sigma^{2}\). What is the probability of drawing two white balls in part (b)?

Let \(X\) be a random variable with \(E(X)=\mu\) and \(V(X)=\sigma^{2}\). Show that the function \(f(x)\) defined by $$f(x)=\sum_{\omega}(X(\omega)-x)^{2} p(\omega)$$ has its minimum value when \(x=\mu\).

A royal family has children until it has a boy or until it has three children, whichever comes first. Assume that each child is a boy with probability \(1 / 2\). Find the expected number of boys in this royal family and the expected number of girls.
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