Continuous Random Variables

Show that $b = 1 - a$.

Given that $E(X) = 1.2$, find the value of $a$.

A random variable $T$ has probability density function given by

Find the value of $c$ such that $P(-c < t < c) = \frac{1}{2}$.

Show that $\text{P}(X < \frac{1}{2}) = \frac{1}{2} + \frac{1}{\pi}$.

Show that $\text{E}(X) = \frac{1}{2} - \frac{2}{\pi^2}$.

Show that $k = \frac{3}{a}$.

It is given that $\text{E}(X) = 1$.

Find the value of $a$.

Find the median of $X$.

Show that $a = \frac{2}{b^2}$.

Show that $P(X < E(X)) = \frac{4}{9}$.

Find the value of $k$.

Find the value of $E(X)$.

Find the probability that a randomly chosen student takes longer than 4.5 minutes to complete the test.

Write down the median of $X$.

Without performing an integration, use your answer to part (a) to find $P(3.5 < X < 4.5)$.

Show that $k = \frac{3}{4}$.

Write down the value of $\mathrm{P}(X \leqslant m)$.

Hence find $\mathrm{P}(\mathrm{E}(X) \leqslant X \leqslant m)$.

Find the probability that a randomly chosen student takes longer than 4.5 minutes to complete the test.

Write down the median of $X$.

Without performing an integration, use your answer to part (a) to find $P(3.5 < X < 4.5)$.

Find an expression for $b$ in terms of $a$.

Given that $\mathrm{E}(X) = \frac{4}{9}$ find the value of $a$.

Using the value of $a$ found in part (b) find the value of $k$ such that $\mathrm{P}(X < k) = \frac{3}{4}$.

Using only this information show that $\text{P}(X > -1) = \frac{245}{256}$.

It is now given that, for $x$ in a suitable domain,

where $k$ is a constant.

Find the value of $k$.

A different random variable $X$ has probability density function $\text{g}(x) = \frac{2}{9}(2 + x - x^2)$. The domain of $X$ is all values of $x$ for which $\text{g}(x) \geqslant 0$.

Find $\text{Var}(X)$.