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Continuous Random Variables

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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