Continuous random variables: probability density functions, expectation, variance

What is a Continuous Random Variable?

A continuous random variable can take any value within a range (often all real numbers). Think of it like measuring the exact height of a plant: you could get 10.3 cm, 10.31 cm, 10.311 cm, and so on—there are infinitely many possibilities.

Probability Density Function (PDF)

For a continuous variable \(X\), the PDF \(f(x)\) gives the relative likelihood of \(X\) being near a particular value.

• Area under the curve over an interval \([a,b]\) equals the probability that \(X\) falls in that interval: $$P(a \le X \le b)=\int_a^b f(x)\,dx$$
• Two key properties:
  1. \(f(x)\ge 0\) for all \(x\)
  2. \(\displaystyle\int_{-\infty}^{\infty} f(x)\,dx = 1\)

📌 Analogy: Imagine a smooth hill. The height of the hill at each point represents how likely you are to land there if you roll a ball down the hill. The total area of the hill is always 1, just like total probability.

Example: Uniform Distribution on \([0,1]\)

The simplest PDF is the uniform distribution where every value in \([0,1]\) is equally likely.

Check: \(\displaystyle\int_0^1 1\,dx = 1\). ??

Expectation (Mean) of a Continuous Variable

The average value you would expect if you could repeat the experiment infinitely many times.

Formula: $$\displaystyle\mu = E[X] = \int_{-\infty}^{\infty} x\,f(x)\,dx$$

📌 Analogy: If you have a jar of marbles of different weights, the expectation is the weight you would get if you could weigh a marble from the jar a million times and average all the weights.

Example (Uniform \([0,1]\)):

$$E[X] = \int_0^1 x \cdot 1\,dx = \left[\frac{x^2}{2}\right]_0^1 = \frac12$$

Variance and Standard Deviation

Variance measures how spread out the values are around the mean.

Formula: $$\displaystyle\sigma^2 = \operatorname{Var}(X) = \int_{-\infty}^{\infty} (x-\mu)^2\,f(x)\,dx$$

Standard deviation is the square root of variance: $$\sigma = \sqrt{\operatorname{Var}(X)}$$

📌 Analogy: Think of a classroom where everyone's height is measured. Variance tells you how much the heights differ from the average height.

Example (Uniform \([0,1]\)):

$$\operatorname{Var}(X) = \int_0^1 (x-\tfrac12)^2\,dx = \frac{1}{12} \approx 0.0833$$ $$\sigma = \sqrt{\tfrac{1}{12}} \approx 0.289$$

Normal Distribution (The “Bell Curve”)

One of the most important PDFs in statistics.

PDF: $$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp\!\Bigl(-\frac{(x-\mu)^2}{2\sigma^2}\Bigr)$$

Key facts:

• Mean \(=\mu\)
• Variance \(=\sigma^2\)
• About 68 % of the area lies within \(\mu\pm\sigma\)
• About 95 % lies within \(\mu\pm2\sigma\)
• About 99.7 % lies within \(\mu\pm3\sigma\)

📌 Exam tip: When a question asks for “the probability that a normally distributed variable lies between two values”, remember to standardise using \(Z=\frac{X-\mu}{\sigma}\) and use Z‑tables or calculator functions.

Exam Tips & Quick Checks

• Always check that your PDF integrates to 1.
• For expectation, if you can’t integrate directly, use symmetry or known results (e.g., for a uniform distribution on \([a,b]\), \(E[X] = \frac{a+b}{2}\)).
• Variance can also be found using \( \operatorname{Var}(X) = E[X^2] - (E[X])^2 \).
• When given a piecewise PDF, split the integral at the points where the function changes.
• Remember that for any continuous variable, \(P(X=x)=0\). Probabilities are always for intervals.
• Use emojis or colour coding in your notes to remember key formulas: e.g., green for expectation, blue for variance.

💡 Practice Question: A continuous random variable \(X\) has PDF $$f(x)=\begin{cases} 2x & \text{if } 0\le x\le1\\ 0 & \text{otherwise} \end{cases}$$ Find \(E[X]\) and \(\operatorname{Var}(X)\).

Solution sketch: \(E[X] = \int_0^1 2x^2\,dx = \frac{2}{3}\). \(E[X^2] = \int_0^1 2x^3\,dx = \frac{1}{2}\). \(\operatorname{Var}(X) = \frac{1}{2} - \left(\frac{2}{3}\right)^2 = \frac{1}{18}\).