Question

Project: Random variables are all around us, from the time we require to commute to school, to the percentage of lecture material we remember for the exam, we can describe much of the world around us using probability. Project Statement: Find a random variable in your day-to-day life, call it X(ω), and do the following:

• Describe X as either quantitative, qualitative, discrete, continuous, etc.

• Give the support of X (i.e. its possible range of values)

• Speculate on its distribution. Is it normal, geometric, exponential, etc. Give specific reasons and justification for this speculation!

• Sample this random variable at least 5 times. • Use this sample to estimate its parameters.

• Give the newly parameterized distribution explicitly.

Answer 1

Random variable-

An example of a random variable in our daily life is time expended in study.

Type of random variable-

Suppose, random variable X denotes time expended (in hours) in study in a day.

Then X is a continuous random variable.

Support of X-

X can take any value in the range [0, 24].

Type of distribution-

Random variable X follows normal distribution.

The causes of such conclusion is as follows.

• Our random variable is of continuous type as normal variate.
• In normal distribution, probability is symmetric about mean and gradually decreases from mean. In case of our proposed random variable, probability is more or less symmetric about its mean and gradually decreases from mean.

Sample values-

Based on previous e days, time expended in study are 12.3, 11.5, 12.0, 11.8 and 11.9 hours.

Estimation of parameters-

Distribution-

We found sample mean as 11.9 and sample variance as 0.085.

So, our distribution is as follows.