The random variable $$$X$$$ takes values in the range $$$1 \leqslant x \leqslant p$$$, where $$$p$$$ is a constant. The graph of the probability density function of $$$X$$$ is shown in the diagram. Find $$$\mathrm{E}(X)$$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_s21_qp_62 Year：2021 Question No：3(b)

Answer:

Gradient $=2$
equation of line is $y=2 x+c$
line passes through $(1,0)$, hence $c=-2$
$y=2 x-2$
$2 \int_{1}^{2}\left(x^{2}-x\right) d x$
$\left.2\left[\frac{x^{3}}{3}-\frac{x^{2}}{2}\right]\right]_{1}^{2}$
$\frac{5}{3}$ or $1.67(3 \mathrm{sf})$

Knowledge points:

6.3.2 use a probability density function to solve problems involving probabilities, and to calculate the mean and variance of a distribution. (Including location of the median or other percentiles of a distribution by direct consideration of an area using the density function.) (Explicit knowledge of the cumulative distribution function is not included.)

Answer:

Knowledge points:

Solution:

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