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Continuous random variables

Define a probability density function (pdf); calculate probabilities as definite integrals; find the mean, median and mode of a continuous distribution; work with uniform and simple polynomial pdfs.

Worked examples

Verifying a pdf and finding a probability

Straightforward

Problem

A random variable XX has f(x)=34(1−x2)f(x) = \dfrac{3}{4}(1 - x^2) for −1≤x≤1-1 \le x \le 1 and f(x)=0f(x) = 0 elsewhere. (a) Verify that ff is a valid pdf. (b) Find P(0≤X≤0.5)P(0 \le X \le 0.5).

Mean and median of a continuous distribution

Moderate

Problem

XX has pdf f(x)=x8f(x) = \dfrac{x}{8} for 0≤x≤40 \le x \le 4. (a) Find E[X]E[X]. (b) Find the median mm.

Finding kk and calculating a probability

Challenging

Problem

XX has pdf f(x)=k(2x−x2)f(x) = k(2x - x^2) for 0≤x≤20 \le x \le 2. (a) Find kk. (b) Find P(X>1)P(X > 1).

Practise

Q1·Straightforward
A continuous random variable XX has probability density function f(x)=kxf(x) = kx for 0≤x≤20 \le x \le 2, and f(x)=0f(x) = 0 otherwise.

Find the value of kk so that ff is a valid pdf.
Q2·Straightforward
XX is uniformly distributed on the interval [3,9][3, 9], so f(x)=16f(x) = \dfrac{1}{6} for 3≤x≤93 \le x \le 9.

Find P(4≤X≤7)P(4 \le X \le 7).
Q3·Straightforward
XX has pdf f(x)=3x2f(x) = 3x^2 for 0≤x≤10 \le x \le 1 (and 00 elsewhere).

Find P(X≤0.5)P(X \le 0.5).
Q4·Straightforward
XX has pdf f(x)=3x2f(x) = 3x^2 for 0≤x≤10 \le x \le 1.

Find the mode of XX (the value of xx at which f(x)f(x) is largest).
Q5·Moderate
XX has pdf f(x)=x8f(x) = \dfrac{x}{8} for 0≤x≤40 \le x \le 4 (and 00 elsewhere).

Find P(1≤X≤3)P(1 \le X \le 3).
Q6·Moderate
XX is uniformly distributed on [0,4][0, 4], with pdf f(x)=14f(x) = \dfrac{1}{4} for 0≤x≤40 \le x \le 4.

Find the mean E[X]E[X].
Q7·Moderate
XX has pdf f(x)=3x2f(x) = 3x^2 for 0≤x≤10 \le x \le 1.

Find the mean E[X]E[X]. Express your answer as a decimal.
Q8·Moderate
XX has pdf f(x)=2(1−x)f(x) = 2(1 - x) for 0≤x≤10 \le x \le 1.

Find the median of XX, rounded to 4 decimal places.

(The median mm satisfies ∫0mf(x) dx=0.5\displaystyle\int_0^m f(x)\,dx = 0.5.)
Q9·Challenging
XX has pdf f(x)=cx(4−x)f(x) = cx(4 - x) for 0≤x≤40 \le x \le 4, and f(x)=0f(x) = 0 otherwise.

Find the value of cc. Express your answer as a fraction in the form p/qp/q.
/
Q10·Challenging
XX has pdf f(x)=332x(4−x)f(x) = \dfrac{3}{32}x(4 - x) for 0≤x≤40 \le x \le 4.

Find E[X]E[X].
Q11·Challenging
XX has pdf f(x)=6x(1−x)f(x) = 6x(1 - x) for 0≤x≤10 \le x \le 1.

Find P ⁣(X≤13)P\!\left(X \le \dfrac{1}{3}\right). Express your answer as a fraction p/qp/q in simplest form.
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Q12·Challenging
XX has pdf f(x)=6x(1−x)f(x) = 6x(1 - x) for 0≤x≤10 \le x \le 1.

Find the mean E[X]E[X].