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The reading given by a thermometer calibrated in ice water
Chapter 2, Problem 8SE(choose chapter or problem)
The reading given by a thermometer calibrated in ice water (actual temperature \(0^\circ \mathrm C\)) is a random variable with probability density function
\(f(x)= \begin{cases}k\left(1-x^{2}\right) & -1<x<1 \\ 0 & \text { otherwise }\end{cases}\)
where k is a constant.
a. Find the value of k.
b. What is the probability that the thermometer reads above \(0^\circ \mathrm C\)?
c. What is the probability that the reading is within \(25^\circ \mathrm C\) of the actual temperature?
d. What is the mean reading?
e. What is the median reading?
f. What is the standard deviation?
Questions & Answers
QUESTION:
The reading given by a thermometer calibrated in ice water (actual temperature \(0^\circ \mathrm C\)) is a random variable with probability density function
\(f(x)= \begin{cases}k\left(1-x^{2}\right) & -1<x<1 \\ 0 & \text { otherwise }\end{cases}\)
where k is a constant.
a. Find the value of k.
b. What is the probability that the thermometer reads above \(0^\circ \mathrm C\)?
c. What is the probability that the reading is within \(25^\circ \mathrm C\) of the actual temperature?
d. What is the mean reading?
e. What is the median reading?
f. What is the standard deviation?
ANSWER:Step 1 of 6
a) Here we have to find the integration of the given pdf And that will be equal to total probability one
\(\begin{array}{l}\int_{-1}^1f(x)dx=1\\ \int_{-1}^1k\left(1-x^2\right)dx=1\\ \left[x-\frac{x^3}{3}\right]_{-1}^1=1/k\\ (1-1/3)-(-1+1/3)=1/\mathrm{k}\\ 2-(2/3)=1/\mathrm{k}\\ \mathrm{k}=3/4\\ \mathrm{k}=0.75\end{array}\)