Continuous Random Variables Questions and Answers

A continuous random variable X has probability density function $f(x) = \dfrac{3}{8}x^2$ for $0 \leq x \leq 2$. Find the value of the cumulative distribution function $F(1)$.

A continuous random variable $X$ has probability density function $f(x) = \dfrac{x}{12}$ for $0 \le x < 3$, $f(x) = k(x-8)$ for $3 \le x \le 8$, and $f(x) = 0$ otherwise, where $k$ is a constant. Since the total area under $f$ must equal 1, what is the value of $k$?

A continuous random variable $X$ has probability density function $f(x) = \dfrac{x}{12}$ for $0 \le x < 3$ and $f(x) = -\dfrac{1}{20}(x-8)$ for $3 \le x \le 8$ (zero otherwise). What is $E(X)$, correct to 2 decimal places?

A continuous random variable $X$ has probability density function $f(x) = \dfrac{x}{12}$ for $0 \le x < 3$ and $f(x) = -\dfrac{1}{20}(x-8)$ for $3 \le x \le 8$ (zero otherwise). What is the median of $X$, correct to 2 decimal places?

A continuous random variable X has probability density function f(x) = (3/8)x^2 for 0 <= x <= 2. What is the cumulative distribution function F(x) for 0 <= x <= 2?

A continuous random variable X has cumulative distribution function F(x) = x^3/8 for 0 <= x <= 2. What is the median m of X? Give your answer to 3 decimal places.

A continuous random variable $X$ has probability density function $f(x) = 0.1x + k$ for $4 \leq x \leq 6$, $f(x) = 0.3$ for $6 \leq x \leq 8$, and $f(x) = 0$ otherwise. Find the value of the constant $k$.

A continuous random variable $X$ has probability density function $f(x) = 0.1x - 0.3$ for $4 \leq x \leq 6$, $f(x) = 0.3$ for $6 \leq x \leq 8$, and $f(x) = 0$ otherwise. Find $P(5 \leq X \leq 7)$.

The continuous random variable $X$ has probability density function $f(x)=\frac{1}{28}(6x-4)$ for $2\le x\le4$ (and 0 otherwise). Find $E(X)$, giving your answer as a fraction or correct to 3 decimal places.

A number X is randomly selected from the interval $(-\pi,\ \pi)$, so that X is uniformly distributed over that interval. Which of these is the cumulative distribution function of X?

X is a continuous random variable with probability density function $f(x)$. The density is the curve $y=x^{2}$ for $0\le x\le 1$, followed by the straight line running from the point $(1;\,1)$ down to the point $(h;\,0)$ on the x-axis, and $f(x)=0$ elsewhere.

X is a continuous random variable with probability density function $f(x)$. The density is the curve $y=x^{2}$ for $0\le x\le 1$, followed by the straight line running from the point $(1;\,1)$ down to the point $(h;\,0)$ on the x-axis, and $f(x)=0$ elsewhere.