Math Problem Statement

Find c for the given pdfs

Find c for the given probability density functions (pdfs)

Solution

To determine the value of cc for the given probability density functions (pdfs), we need to ensure that each function satisfies the properties of a valid pdf. Specifically, the total area under the curve of the pdf must equal 1. This translates mathematically to:

abf(x)dx=1\int_{a}^{b} f(x) \, dx = 1

For each case in the image:

(a) f(x)=cx3f(x) = c x^3 for 0x10 \leq x \leq 1

We need to solve the following integral: 01cx3dx=1\int_0^1 c x^3 \, dx = 1 Solving this: c01x3dx=1c \int_0^1 x^3 \, dx = 1 c[x44]01=1c \left[ \frac{x^4}{4} \right]_0^1 = 1 c×14=1c \times \frac{1}{4} = 1 c=4c = 4

(b) f(x)=cf(x) = c for 4x4-4 \leq x \leq 4

We need to solve: 44cdx=1\int_{-4}^{4} c \, dx = 1 Solving this: c×(4(4))=1c \times (4 - (-4)) = 1 c×8=1c \times 8 = 1 c=18c = \frac{1}{8}

(c) f(x)=c(2x)f(x) = c \cdot (2 - x) for 0x20 \leq x \leq 2

We need to solve: 02c(2x)dx=1\int_0^2 c (2 - x) \, dx = 1 Solving this: c02(2x)dx=1c \int_0^2 (2 - x) \, dx = 1 First, compute the integral: [ \int_0^2 (2 - x) , dx = \left[ 2x - \frac{x^2}{2} \right]_0^2 = \left[ 4 - 2 \right] = 2 ] So, we get: c×2=1c \times 2 = 1 c=12c = \frac{1}{2}

Let me know if you'd like further details on any step or additional questions! Here are five related questions for practice:

  1. What is the general condition for a function to be a valid probability density function?
  2. How would you normalize a pdf if it’s not normalized already?
  3. How would the pdf change if the range for (b) was 2x2-2 \leq x \leq 2?
  4. What happens to the constant cc in a pdf if the range of integration is doubled?
  5. How do you interpret the value of cc in the context of these problems?

Tip: Always ensure that the integral of any pdf over its range equals 1 to satisfy the normalization condition!

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Math Problem Analysis

Mathematical Concepts

Probability Density Functions
Integral Calculus
Normalization Condition

Formulas

∫ f(x) dx = 1 (Normalization condition for a valid pdf)
∫ c x^3 dx
∫ c dx
∫ c (2 - x) dx

Theorems

Fundamental Theorem of Calculus
Normalization Condition for PDFs

Suitable Grade Level

University Level (Advanced Calculus/Probability)