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Chapter 12: Problem 8

Find the value of \(k\) that makes the given function a probability density function on the specified interval. \(f(x)=k x^{2}, 0 \leq x \leq 2\)

Short Answer

Expert verified
k=\frac{9}{16}

Step by step solution

01

Understand the definition of a probability density function

A probability density function (PDF) must integrate to 1 over its interval. For the given function, we need to find the value of k such that the integral of f(x) from 0 to 2 equals 1.
02

Set up the integral of the function

Set up the integral:
03

Perform the integration

Integrate the function with respect to x:
04

Solve for k

Set the integral equal to 1 and solve for k:
05

Verify the solution

Substitute the value of k back into the function and verify that it integrates to 1 over the interval [0, 2].

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Key Concepts

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

Integration
Integration is the process of finding the integral of a function. In this exercise, we use integration to ensure that our probability density function sums up to 1.
For the function \(f(x) = k x^2\) over the interval \[0, 2\], we set up the integral to find \(k\).
We perform the integration of \(f(x) = k x^2\) from 0 to 2:
\[\int_0^2 kx^2 \ dx\]
This will help us find the value of \(k\) that normalizes the function.
Calculus
Calculus plays a big role in understanding how functions behave and how we can manipulate them. In our exercise, we need to use basic calculus to integrate the function and find the unknown constant \(k\).
The integral we need to solve is:
\[\int_0^2 k x^2 \ dx\]
This involves using the power rule of integration. The result of this integration gives us an equation that helps us solve for \(k\).
Solving for Constants
After setting up the integral, we solve for the constant \(k\). We integrate \(kx^2\) over the interval \[0, 2\]:
\[\int_0^2 k x^2 \ dx = k \left[ \frac{x^3}{3} \right]_0^2 \]
Which simplifies to:
\[k \left( \frac{8}{3} - 0 \right) = 1\]Now, solve for \(k\):
\[k \cdot \frac{8}{3} = 1\]
\[k = \frac{3}{8}\]
This value of \(k\) makes our function a valid probability density function.
Verifying Solutions
After calculating \(k\), it's important to verify our solution to ensure the function is a valid probability density function. We substitute \(k = \frac{3}{8}\) back into the integral and check if the result equals 1:
\[\int_0^2 \frac{3}{8} x^2 \ dx\]
Performing the integration:
\[\frac{3}{8} \int_0^2 x^2 \ dx\]
\[\frac{3}{8} \left[ \frac{x^3}{3} \right]_0^2\]
This simplifies to:
\[\frac{3}{8} \cdot \frac{8}{3} = 1\]
Thus, the integration confirms that our function is properly normalized.

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Most popular questions from this chapter

In a certain cell population, cells divide every 10 days, and the age of a cell selected at random is a random variable \(X\) with the density function \(f(x)=2 k e^{-k x}, 0 \leq x \leq 10, k=(\ln 2) / 10\). A random variable \(X\) has a density function \(f(x)=\frac{2}{3} x\) on \(1 \leq x \leq 2\). Find \(a\) such that \(\operatorname{Pr}(a \leq X)=\frac{1}{3}\).

In a certain cell population, cells divide every 10 days, and the age of a cell selected at random is a random variable \(X\) with the density function \(f(x)=2 k e^{-k x}, 0 \leq x \leq 10, k=(\ln 2) / 10\). Upon examination of a slide, \(10 \%\) of the cells are found to be undergoing mitosis (a change in the cell leading to division). Compute the length of time required for mitosis; that is, find the number \(M\) such that $$ \int_{10-M}^{10} 2 k e^{-k x} d x=.10 $$

The cumulative distribution function for a random variable \(X\) on the interval \(1 \leq x \leq 2\) is \(F(x)=\frac{4}{3}-4 /\left(3 x^{2}\right)\). Find the corresponding density function.

Number of Cars at a Tollgate During a certain part of the day, an average of five automobiles arrives every minute at the tollgate on a turnpike. Let \(X\) be the number of automobiles that arrive in any 1 -minute interval selected at random. Let \(Y\) be the interarrival time between any two successive arrivals. (The average interarrival time is \(\frac{1}{5}\) minute.) Assume that \(X\) is a Poisson random variable and that \(Y\) is an exponential random variable. (a) Find the probability that at least five cars arrive during a given 1-minute interval. (b) Find the probability that the time between any two successive cars is less than \(\frac{1}{5}\) minute.

In Exercises \(7-12\), find the value of \(k\) that makes the given function a probability density function on the specified interval. \(f(x)=k x, 1 \leq x \leq 3\)
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