37. In certain experiment the error made in determining the density of a substance is a random

variable having Uniform density with α = -0.015 and β=0.015. Find the probability that

(i) such an error will be between -0.01 and 0.02. and (ii) such an error will exceed 0.005.

Given the problem, the error in determining the density of a substance is uniformly distributed between \(\alpha = -0.015\) and \(\beta = 0.015\). 

For a uniform distribution \( U(\alpha, \beta) \), the probability density function (PDF) is given by:

\[
f(x) = \frac{1}{\beta - \alpha}, \quad \text{for } \alpha \leq x \leq \beta
\]

Here, \(\alpha = -0.015\) and \(\beta = 0.015\), so:

\[
f(x) = \frac{1}{0.015 - (-0.015)} = \frac{1}{0.03} = \frac{1}{0.03} = 33.33
\]

Now, let's calculate the probabilities:

### (i) Probability that the error is between -0.01 and 0.02
We need to find \( P(-0.01 \leq X \leq 0.02) \).

Since \(X\) is uniformly distributed:

\[
P(a \leq X \leq b) = \int_a^b f(x) \, dx = f(x) \cdot (b - a)
\]

For \(a = -0.01\) and \(b = 0.02\):

\[
P(-0.01 \leq X \leq 0.02) = f(x) \cdot (0.02 - (-0.01)) = 33.33 \times 0.03 = 1
\]

\[
P(-0.01 \leq X \leq 0.02) = 1
\]

### (ii) Probability that the error will exceed 0.005
We need to find \( P(X > 0.005) \).

\[
P(X > 0.005) = 1 - P(X \leq 0.005)
\]

We calculate \( P(X \leq 0.005) \):

\[
P(X \leq 0.005) = \int_{-0.015}^{0.005} f(x) \, dx = f(x) \cdot (0.005 - (-0.015)) = 33.33 \times 0.02 = 0.6667
\]

Thus:

\[
P(X > 0.005) = 1 - 0.6667 = 0.3333
\]

### Summary of Results
1. \( P(-0.01 \leq X \leq 0.02) = 1 \)
2. \( P(X > 0.005) = 0.3333 \)

Would you like more details on these calculations or have any other questions? Here are some related questions to consider:

1. What is the cumulative distribution function (CDF) for a uniform distribution?
2. How do you calculate the mean of a uniform distribution?
3. How does the probability change if the distribution range is modified?
4. What is the variance of a uniform distribution?
5. How does uniform distribution apply to real-world scenarios?

**Tip:** For uniform distributions, the probability of any subinterval is directly proportional to the length of that interval relative to the entire range.

Explore how to calculate probabilities for errors in density measurement using a uniform distribution with specific alpha and beta values. Understand the PDF and CDF concepts with detailed examples.

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