Continuous Random Variables [Video Tutorial]

This tutorial covers material encountered in chapter 15 of the VCE Mathematical Methods Textbook, namely:

• Continuous random variables
• Probability density functions
• Mean, variance and standard deviation of a continuous random variable
• Interquartile range and median of a continuous random variable

Q1 – Probability Density Function, Variance and Standard Deviation of a Continuous Random Variable

Q2 – Probability Density Function and Median of a Continuous Random Variable

Q3 – Variance, Standard Deviation and Interquartile Range of a Continuous Random Variable

Q4 – Application of Continuous Random Variable – Interval Calculation

Q5 – Application of Continuous Random Variable – Find the Expectation

Worksheet

Q1. The probability density function of a random variable \(X\) has \(E(X)=\dfrac{1}{3}\) and is given by:

\( f(x) = \begin{cases} a+bx^2, & \text{for } 0\leq x\leq 1\\ 0, & \text{otherwise }\\ \end{cases} \)

where \(a,\,b\,\in\R\), find:

(a) \(a\) and \(b\)

(b) \(Var(X)\) and \(sd(X)\)

(c) \(P(X<\frac{1}{2})\)

(d) \(P(X>\frac{1}{4})\)

(e) \(P(\frac{1}{4}<X<1\,|\,X<\frac{1}{2}) \)

Q2. The probability density function of a random variable $X$ is given by:

\( f(x) = \begin{cases} \dfrac{\sin(x)}{a}, & \text{for } 0\leq x\leq \dfrac{\pi}{2}\\ 0, & \text{otherwise }\\ \end{cases} \)

where \(a\,\in\R\)

(a) Find \(a\)

(b) Find \(m\), the median of \(X\)

(c) Find \(P\left(X<\dfrac{\pi}{3}\right)\)

Q3. The probability density function of a random variable $X$ is given by:

\( f(x) = \begin{cases} \dfrac{1}{x}, & \text{for } 1\leq x\leq e\\ 0, & \text{otherwise }\\ \end{cases} \)

(a) Find \(\mu (X)\), and \(\sigma^2(X)\)

(b) Find the interquartile range of \(X\)

Q4. The weight, \(X\) grams, of one of Robert’s chicken and chive dumplings is a continuous random variable. If the average weight of a dumpling is 25 grams and the standard deviation is 4 grams, find an approximate interval for the weight of 95% of the dumplings

Q5. The amount of marinara sauce used each day at an Hadliegh’s Pizza Parlour is a continuous random variable \(X\) with a mean of 2 tonnes. The cost, \(C\) dollars, to make the sauce is \(C=50+200X\). Find \(E(C)\).

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