A continuous random variable X that can assume values between 0 and 2 has a density function given by:
\[ \begin{align}
f(x) &= \frac{x}{2} \quad 0 \le x \le 2 \\
&= 0 \qquad \text{elsewhere}
\end{align}\]
(a) Confirm this is a probability distribution.
(b) What is the probability of X taking a value less than 1?
Solve
A continuous random variable X that can assume values between 0 and 1 has a density function given by:
\[ \begin{align}
f(x) &= 2(1 - x) \quad 0 \le x \le 1 \\
&= 0 \qquad \text{elsewhere}
\end{align}\]
(a) Confirm this is a probability distribution.
(b) What is the probability of X taking a value less than 1/2?
Solve
A continuous random variable X has a density function given by:
\[ \begin{align}
f(x) &= 2e^{-2x} \qquad x \ge 0 \\
&= 0 \qquad \text{elsewhere}
\end{align}\]
Find the probability of X taking a value between 0 and 3, given it has taken a value less than 4.
Solve
A continuous random variable X that can assume values between 0 and 1 has a cumulative distribution given by:
\[ \begin{align}
F(x) &= \frac{x^2+4x}{5} \quad 0 \le x \le 1 \\
&= 0 \quad x \lt 0 \\
&= 1 \quad x \gt 1
\end{align}\]
(a) Find the corresponding probability density function.
(b) What is the probability of X taking a value less than 0.75?
Solve
A continuous random variable X that can assume values between 0 and 1 has a probability density function given by:
\[ \begin{align}
f(x) &= 2 (1 - x)\quad 0 \le x \le 1 \\
&= 0 \quad \text{elsewhere}
\end{align}\]
(a) Find the corresponding cumulative density function.
(b) What is the probability of X taking a value less than 0.75?
Solve