The pH level, a measure of acidity, is important in studies of acid rain. For a certain lake, baseline measurements of acidity are made so that any changes caused by acid rain can be noted. The pH for water samples from the lake is a random variable X, with probability density function
$$ \begin{aligned} f(x) & =\frac{3}{8}(7-x)^2, 5\leq x\leq 7\\ &= 0, \text{elsewhere} \end{aligned} $$
(a) Sketch the pdf
(b) Find the probability that the pH for a sample of water taken from this lake will be less than 6.
(c) Find the probability that a sample of water taken from this lake will have a pH less than 5.5, given that it is known to have a pH less than 6.
Solution
(a) Sketch of the pdf
(b) The probability that the pH for a sample of water taken from this lake will be less than 6 is
$$ \begin{aligned} P(X< 6) &=\int_5^6 f(x)\; dx\\ &= \int_5^6 \frac{3}{8}(7-x)^2\; dx\\ &= \frac{3}{8} \bigg[\frac{-(7-x)^3}{3}\bigg]_5^6\\ &= \frac{1}{8}\bigg[-(7-6)^3+(7-5)^3\bigg]\\ &= \frac{1}{8}\bigg[-1+8\bigg]\\ &=\frac{7}{8} \end{aligned} $$
(c) The probability that a sample of water taken from this lake will have a pH less than 5.5, given that it is known to have a pH less than 6 is
$$ \begin{aligned} P(X< 5.5|X<6) &=\frac{P(X< 5.5 \cap X< 6)}{P(X<6)}\\ &=\frac{P(X< 5.5)}{P(X<6)}\\ &= \frac{\int_5^{5.5} f(x)\; dx}{\int_5^6 f(x)\; dx}\\ &= \frac{\int_5^{5.5}\frac{3}{8}(7-x)^2\; dx}{7/8}\\ &= \frac{3}{7} \bigg[\frac{-(7-x)^3}{3}\bigg]_5^{5.5}\\ &= \frac{1}{7}\bigg[-(7-5.5)^3+(7-5)^3\bigg]\\ &= \frac{1}{7}\bigg[-(1.5)^3+(2)^3\bigg]\\ &=\frac{1}{7} (-3.375 +8)\\ &= 0.660714 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
