Question

A continuous random variable has probability density function

f(x) = ×/50, for 0 < x < 10, and f(x) = 0 elsewhere Find
P(1.1 < X < 4.2), to an accuracy of 2 decimal places.

Answer 1

To find the probability P(1.1 < X < 4.2) for the given probability density function f(x) = x/50, we integrate it over the given interval and find that the probability is 0.16.

Step-by-step explanation:

Finding the Probability for a Continuous Random Variable

To find P(1.1 < X < 4.2) for the continuous random variable with probability density function (pdf) f(x) = x/50, where 0 < x < 10, we need to integrate the pdf over the interval from 1.1 to 4.2. The calculation for this is:

\[P(1.1 < X < 4.2) = \int_{1.1}^{4.2} \left(\frac{x}{50}\right) dx\]

Perform the integration:

\[= \left. \left(\frac{x^2}{100}\right) \right|_{1.1}^{4.2}\]

\[= \left(\frac{4.2^2}{100}\right) - \left(\frac{1.1^2}{100}\right)\]

\[= \frac{17.64}{100} - \frac{1.21}{100}\]

\[= 0.1764 - 0.0121\]

\[= 0.1643\]

After rounding to two decimal places, we get:

\[P(1.1 < X < 4.2) = 0.16\]

This result tells us that the probability that the random variable X takes on a value between 1.1 and 4.2 is 0.16.