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    • ▾Calculus 2
      • ▸Integration techniques
        • •Intro to calc 2
        • ▸Basic indefinite integrals
          • •Reverse power rule for indefinite integrals
          • •Sum, difference, and reverse power rules for indefinite integrals
        • •Integration by parts
        • ▸Integrals involving trig functions
          • •Lesson
          • •Practice
        • •Trig substitutions
        • •Integration by partial fractions
        • •Integral recurrence relations
        • •Integrating quadratics by completing the square
        • •Integration by long division
        • •Improper integrals
        • •Comparison tests for convergence
        • ▸Approximating definite integrals
          • •Riemann sums
          • ▸Simpson's rule
            • •Practice: Simpson's rule
          • ▸Trapezoidal rule
            • •Lesson
            • •Practice
          • •Midpoint rule
      • ▾Applications of integrals
        • ▸Arc length (calculus 2)
          • •Using the formula
        • ▸Center of mass (calculus 2)
          • •Center of mass for one and two-dimensional systems
          • •Center of mass for planar lamina
          • •Pappus's theorem
        • •Hydrostatic force
        • •Mode of a continuous random variable from a probability density function
        • •Surface area of solids of revolution
        • •Gabriel's Horn
        • •Applications of trigonometric integrals
        • •Integrating exponential functions
        • •Logarithmic functions
      • ▸Parametric equations and polar coordinates
        • •Cycloid area and length
        • •Eliminating the parameter
        • •Testing polar equations for symmetry
        • •Deriving the equations of a cycloid, epicycloid, and hypocycloid
      • ▸Series and sequences
        • •Maclaurin series
        • •Finding the sum of a telescoping series
        • ▸Convergent and divergent geometric series
          • •Lesson
          • •Practice
        • •Alternating harmonic series
        • •Alternating series test
        • •Partial sums
        • •Ratio test
        • ▸Root test
          • •Lesson
          • •Practice
        • •Absolute convergence, conditional convergence, and divergence
        • •Riemann's rearrangement theorem
        • •Taylor series
      • ▸Vectors
        • •Intro to vectors
        • •Length of a vector
        • •Cross product
        • •Intro to the dot product
      • •Competition problems (calculus 2)
     › Calculus 2 › Applications of integrals

    Mode of a continuous random variable from a probability density function

    In this section, you will learn how to find the mode of a continuous random variable from a probability density function. Before attempting this section, you should be very good at finding derivatives and performing elementary algebra.

    Conclude by giving your students these challenges:

    • A Chordingly by NRICH
    • Pumpkin Pie Problem by NRICH
    • Olympic Magic by NRICH
    • Changing Mean

    Challenge from Westdene Primary, Brighton by NRICH

    Here's the solution: Start by computing the first few powers of \(7\) and \(9,\) until a cycle is found for each:

    $$\begin{align} 7^0 = 1 \\ 7^1 = 7 \\ 7^2 = 49 \\ 7^3 = 343 \\ 7^4 = 2401 \\[1em] 9^0 = 1 \\ 9^1 = 9 \\ 9^2 = 81 \end{align}$$

    Notice the last digit, of a power of 7, could be \(1, 7, 9, or 3,\) each equally likely. Also, notice the last digit, of a power of 9, could be \(1\) or \(9,\) each equally likely. By the multiplication principle, there are thus \(4 \cdot 2 = 8\) pairings, \(2\) of which, have a sum whose last digit is \(0,\) or in other words, is a multiple of \(10.\) Thus, the probability that \(7^x + 9^y\) is a multiple of \(10,\) is \(2/8 = 1/4.\)

    YouTube videos

    • 72296Continuous Random Variables: Mode
      MrNichollTV
    • 72297Mode for a Continuous Random Variable | ExamSolutions
      ExamSolutions