State whether the following graphs are probability density functions.
State whether the following functions describe continuous probability distributions:
f \left( x \right)=0.2 for the interval 1 \leq x \leq 6.
f \left( x \right)=\dfrac{x}{12} in the domain \left[0,6 \right]
f \left( x \right) = \begin{cases} \dfrac{x^3}{324} & \text{for } 0 \leq x \leq 3 \\ 0 & \text{for all other }x \end{cases}
f \left( x \right)=\dfrac{x^2}{21} in the domain 1 \leq x \leq 4
f \left( x \right) = \begin{cases} \dfrac{x}{8} & \text{for } 1 \leq x \leq 8 \\ 0 & \text{for all other }x \end{cases}
Consider the probability density function p \left( x \right) drawn for a random variable X:
Find the area between p \left( x \right) and the x axis.
State the features of p \left( x \right) which are also features of all probability distribution functions.
Calculate P \left(X \leq 54 \right).
Calculate P \left(X \gt 34 \right).
Calculate P \left(44 \lt X \leq 53 \right).
Calculate P \left(X \leq 56 | X \geq 44 \right) .
Consider the probability density function p \left( x \right) = \dfrac{1}{40} for 60 \leq x \leq 100 and p \left( x \right) = 0 otherwise.
Sketch the graph of the probability density function p \left(x \right).
Calculate P \left( X \gt 80 \right).
Calculate P \left(X \leq 65 | X \leq 90 \right).
Calculate P \left(X \leq 70 \cap X \geq 80 \right).
Consider the probability density function p \left( x \right) drawn for a random variable X:
Find the area between p \left( x \right) and the \\ x-axis.
State features of p \left( x \right) which are also features of all continuous probability distribution functions.
Calculate P \left(X \lt 3 \right).
Calculate P \left(X \gt 3 | X \leq 4 \right).
Consider the probability density function p \left( x \right) drawn for a random variable X:
Find the area between p \left( x \right) and the x-axis.
State the features of p \left( x \right) which are also features of all continuous probability distribution functions.
Calculate P \left (\dfrac{11}{2} \leq X \lt \dfrac{17}{2} \right).
Calculate P \left (X \lt \dfrac{17}{2}|X \geq \dfrac{11}{2} \right).
Consider the probability density function p \left( x \right) drawn for a random variable X:
Find the area between p \left( x \right) and the \\ x-axis.
State the features of p \left( x \right) which are also features of all continuous probability distribution functions.
State the probability density function for X.
Calculate P \left(X \lt 2|X \gt 1 \right).
Consider the probability density function p \left( x \right) drawn for a random variable X:
Find the area between p \left( x \right) and the \\ x-axis.
State the features of p \left( x \right) which are also features of all probability distribution functions.
State the probability density function for X.
Calculate P \left(X \geq 4 \right).
Calculate P \left(X \gt 3|X \lt 4 \right).
Consider the probability density function:
p\left( x \right) = \begin{cases} \dfrac{-x}{40}+\dfrac{1}{5}, & 0 \leq x \leq 8 \\ \dfrac{x}{160}-\dfrac{1}{20}, & 8 \lt x \leq 16 \\ 0, & \text{otherwise} \end{cases}
Sketch the graph of the probability density function p\left( x \right).
Calculate the area under p \left( x \right).
Explain the reason why p \left( x \right) is a probability density function.
The probability density function of a random variable X is shown. Its non-zero values lie in the region 0 \leq x \leq k:
Calculate the value of k.
State the equation that defines the probability distribution function of X in the domain 0 \leq x \leq k.
Calculate P \left(X \lt 0.8 k \right) to two decimal places.
Calculate P \left(X \geq 0.2 k \right) to two decimal places.
Calculate P \left(X \lt 7 | X \gt 2 \right).
Let f \left(x \right)=\dfrac{3}{4} \left(x^2-4x+3 \right) be a function defined on the closed interval \left[0,4 \right].
Calculate \int_0^4 f \left(x \right)dx.
Sketch the graph of y=f \left(x \right).
State whether f \left(x \right) is a PDF.
A probability density function has the equation f \left(x \right)=\dfrac{x^4}{3355} over the domain \left[2,b \right].
Find the value of b.
Consider the continuous probability distribution f \left(x \right)=\begin{cases} kx^3 & \text{for } 0 \leq x \leq 5 \\ 0 & \text{for all other } x \end{cases}.
Find the value of k.
A function is given by f \left(x \right)=\dfrac{x^2}{72}. Find the domain of this probability density function starting at x=0.
A PDF is given by f \left(x \right)=\dfrac{2x^5}{87\,381} over the interval 1 \leq x \leq b. Find the value of b.
The continuous probability distribution is defined by f \left(x \right)=ax^2 on the interval 0 \leq x \leq 5.
Find the value of a.
Calculate:
The continuous random variable X has the PDF shown:
Find the value of a.
Calculate:
A continuous probability function is given by f \left(x \right)=ke^x, defined on the domain \left[1,6 \right].
Find the exact value of k.
Calculate:
Consider y=\sin x.
Is y=\sin x a PDF on the domain \left[0, \dfrac{\pi}{2}\right]?
Calculate:
The mass of a 6 week old puppy, in grams, is modeled by a continuous random variable X which has probability density function p \left( x \right) defined by \\ p \left( x \right)=\begin{cases} k \sin \left( \dfrac{\pi}{240} \left(x - 450\right)\right) & \text{for } 450 \leq x \leq 690 \\ 0 & \text{otherwise} \end{cases}
Find the value of k.
Calculate the probability that a randomly chosen puppy weighs less than 550 grams.
Calculate the probability that a randomly chosen 6 week puppy weighs more than 570 grams, if we are told that it weighs less than 630 grams.