Q.A discrete random variable $X$ has the probability distribution: $X = 0,1,2,3$ with $P(X) = k,\ 2k,\ 3k,\ 4k$ respectively. Find the value of $k$.

Birth year takes countable whole-number values, so it is discrete.

A discrete variable assumes only isolated, countable values, whereas a continuous variable can take any value in an interval. Height, weight and blood pressure of a student are measured on a continuous scale, so they are continuous. The birth year can only be a whole number (e.g. 2005, 2006), which is countable, hence discrete.

The maximum temperature during a day is a continuous variable, because it can take any value in an interval — option (c).

GSEB Class-12 Statistics, Random Variable chapter:

• A discrete variable takes isolated (usually whole-number) values obtained by counting.
• A continuous variable can take any value within a range and is obtained by measurement.

Checking the options:

• Number of accidents — a count → discrete.

• Number of rainy days — a count → discrete.

• Number of children in a family — a count → discrete.

• Maximum temperature during a day — a measured value that can be any real number in a range (e.g. $34.6^\circ$C) → continuous.

A discrete random variable takes only countable (finite or countably infinite), isolated values.

A random variable is a real-valued function defined on the sample space of a random experiment. It is called a discrete random variable when the set of values it can assume is countable — either finite or countably infinite — and the values are distinct and separated (e.g. the number of heads in tosses of a coin, the number of defective items in a lot). Between two consecutive possible values it cannot take intermediate values.

Birth year takes only isolated whole-number values, so it is a discrete variable; height, weight and blood pressure are continuous.

Reasoning. A discrete variable can assume only specific separated values (usually counts), while a continuous variable can take any value in an interval. Height, weight and blood pressure can take any value on a continuous scale, but birth year can only be a whole number such as 2004, 2005, … — hence discrete.