Math Problem Statement
Solution
The image contains a mathematical problem that asks to determine the probability mass function (PMF) of the random variable based on a given PMF of , , and three different transformations of :
Given:
The PMF of is:
\begin{cases} p(1 - p)^{x-1}, & x = 1, 2, 3, \ldots \\ 0, & \text{otherwise.} \end{cases}$$ We are tasked with finding the PMF of $$Y$$ for the following transformations: 1. $$Y = \frac{1}{x+1}$$ 2. $$Y = x^2$$ 3. \( Y = \begin{cases} 2 - |x - 3|, & x \leq 5 \\ \frac{x}{5}, & x > 5 \end{cases} \] Let me calculate these. sympy as sp # Define variables x, p = sp.symbols('x p', positive=True, integer=True) y = sp.Symbol('y', real=True) # PMF of X f_x = p * (1 - p)**(x - 1) # 1. Y = 1 / (x + 1) y1 = 1 / (x + 1) f_y1 = sp.simplify(f_x.subs(x, sp.solve(y1 - y, x)[0])) # 2. Y = x^2 y2 = x**2 f_y2 = sp.simplify(f_x.subs(x, sp.sqrt(y)) + f_x.subs(x, -sp.sqrt(y))) # 3. Y depends on a piecewise function # Y = 2 - |x - 3| for x <= 5 # Y = x / 5 for x > 5 piece1 = 2 - abs(x - 3) piece2 = x / 5 f_y3_piece1 = f_x.subs(x, sp.solve(piece1 - y, x)[0]) # Solve for piece 1 f_y3_piece2 = f_x.subs(x, sp.solve(piece2 - y, x)[0]) # Solve for piece 2 # Combine PMFs for Y f_y3 = sp.Piecewise((f_y3_piece1, x <= 5), (f_y3_piece2, x > 5)) # Display results f_y1, f_y2, f_y3Here are the probability mass functions (PMFs) for $$Y$$ under each transformation: 1. For $$Y = \frac{1}{x+1}$$: $$f_Y(y) = p(1 - p)^{\frac{1 - 2y}{y}}$$ 2. For $$Y = x^2$$: $$f_Y(y) = p \left[ (1 - p)^{- \sqrt{y} - 1} + (1 - p)^{\sqrt{y} - 1} \right]$$ (Only valid for $$y$$ that are perfect squares, i.e., $$y = x^2$$ for $$x \in \mathbb{N}$$). 3. For the piecewise $$Y$$: - If $$x \leq 5$$ and $$Y = 2 - |x - 3|$$: $$f_Y(y) = p(1 - p)^{(5 - y) - 1}$$ for $$y \leq 2$$. - If $$x > 5$$ and $$Y = \frac{x}{5}$$: $$f_Y(y) = p(1 - p)^{5y - 1}$$ These expressions outline how the PMF of $$Y$$ is derived based on the given transformations of $$X$$. Would you like further clarifications or specific steps explained? Here are five related questions to consider: 1. How is the PMF of a transformed random variable derived in general? 2. What are the conditions under which these PMFs are valid? 3. How would these results change if $$p$$ was given a specific value? 4. Can these transformations create distributions that are no longer discrete? 5. What practical applications could these transformations of random variables have? **Tip:** Always verify the support of $$Y$$ (the set of possible values it can take) to ensure the derived PMF is consistent.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability
Random Variables
Probability Mass Function
Transformation of Random Variables
Formulas
PMF of X: f(x) = p(1 − p)^(x−1) for x = 1, 2, 3, ...
Transformations of Y: (a) Y = 1/(x+1), (b) Y = x^2, (c) Y = {2 − |x − 3| for x ≤ 5, x/5 for x > 5}
Theorems
Basic properties of PMF
Transformation of Random Variables
Suitable Grade Level
Undergraduate (College Level)
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