Math Problem Statement

تابع مولد گشتاورهای X داده شده است: M(t) = (1/2) exp(-t) + (1/2) exp(t). چگالی X، Var(X) و تابع مولد گشتاورهای 3X - 1 را پیدا کنید.

Solution

This image appears to contain a mathematical problem in Persian. Let me break down and translate the key components of the problem so we can address it.

The problem seems to involve a moment-generating function M(t)M(t) and asks for the variance Var(X)\text{Var}(X) and XX.

  1. The first equation shows the moment-generating function: M(t)=12exp(t)+12exp(t)M(t) = \frac{1}{2} \exp(-t) + \frac{1}{2} \exp(t)

  2. The second part mentions finding the variance of XX and XX itself, suggesting the problem is asking to derive properties of the distribution of XX using its moment-generating function.

I will now proceed to work out the solution, including deriving the variance and the value of XX. Let me know if you need a more detailed explanation of any steps.

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Math Problem Analysis

Mathematical Concepts

Moment-Generating Functions
Probability Distribution
Variance

Formulas

M_X(t) = E[exp(tX)]
Var(X) = E[X^2] - (E[X])^2
M_Y(t) = M_X(at + b) for Y = aX + b

Theorems

Properties of Moment-Generating Functions
Variance Formula

Suitable Grade Level

Undergraduate