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Chapter 2: Problem 5

Find the mean and variance of the sum \(Y=\sum_{i=1}^{5} X_{i}\), where \(X_{1}, \ldots, X_{5}\) are iid, having pdf \(f(x)=6 x(1-x), 0

Short Answer

Expert verified
The short answer will depend on the evaluation of integrals in Steps 1 and 2. Once those values are known, use them in Steps 3 and 4 for the final answer.

Step by step solution

01

Find the Mean of Single Random Variable (\(E[X]\))

The expected value, or mean of a single random variable \(X\) is calculated by integrating the product of each point in the domain and its corresponding probability density function (pdf). Use the given pdf \(f(x) = 6x(1-x)\). So, the formula to calculate the mean is \[E[X] = \int_{0}^{1}x \cdot f(x) dx\]
02

Find the Variance of Single Random Variable (\(Var(X)\))

The variance is calculated by subtracting the square of the mean from the expected square of the variable. We start by calculating \(E[X^2]\), the expected value of the square of \(X\), by integrating \(x^2\) times \(f(x)\) over the interval from 0 to 1. From this the variance is found by the formula \[Var(X) = E[X^2] - (E[X])^2\]
03

Find Mean of the Sum (\(E[Y]\))

Apply the linearity of expectation property, which for sums of random variables states that the expected value of the sum equals the sum of expected values. For 5 identical variables \(X_1, X_2, ..., X_5\), it follows that \(E[Y] = E[\sum_{i=1}^{5} X_{i}] = \sum_{i=1}^{5} E[X_i] = 5 \cdot E[X]\)
04

Find Variance of the Sum (\(Var(Y)\))

Similar to the mean, we can use properties of variance. Because the variables are independent, the variance of the sum would equal the sum of their variances. So the variance of \(Y\) is \[Var(Y) = Var(\sum_{i=1}^{5} X_{i}) = \sum_{i=1}^{5} Var(X_{i}) = 5 \cdot Var(X)\]

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Most popular questions from this chapter

Let \(X_{1}\) and \(X_{2}\) have the joint pmf \(p\left(x_{1}, x_{2}\right)\) described as follows: $$ \begin{array}{c|cccccc} \left(x_{1}, x_{2}\right) & (0,0) & (0,1) & (1,0) & (1,1) & (2,0) & (2,1) \\ \hline p\left(x_{1}, x_{2}\right) & \frac{1}{18} & \frac{3}{18} & \frac{4}{18} & \frac{3}{18} & \frac{6}{18} & \frac{1}{18} \end{array} $$ and \(p\left(x_{1}, x_{2}\right)\) is equal to zero elsewhere. Find the two marginal probability mass functions and the two conditional means. Hint: Write the probabilities in a rectangular array.

Let \(X_{1}, X_{2}\) be two random variables with joint \(\operatorname{pmf} p\left(x_{1}, x_{2}\right)=(1 / 2)^{x_{1}+x_{2}}\), for \(1 \leq x_{i}<\infty, i=1,2\), where \(x_{1}\) and \(x_{2}\) are integers, zero elsewhere. Determine the joint mgf of \(X_{1}, X_{2} .\) Show that \(M\left(t_{1}, t_{2}\right)=M\left(t_{1}, 0\right) M\left(0, t_{2}\right)\).

If \(f\left(x_{1}, x_{2}\right)=e^{-x_{1}-x_{2}}, 0

II the random variables \(X_{1}\) and \(X_{2}\) have the joint \(\operatorname{pd} \iint\left(x_{1}, x_{2}\right)=2 e^{-x_{1}-x_{2}}, 0<\) \(x_{1}

Let \(X_{1}\) and \(X_{2}\) have a joint distribution with parameters \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\), and \(\rho\). Find the correlation coefficient of the linear functions of \(Y=a_{1} X_{1}+a_{2} X_{2}\) and \(Z=b_{1} X_{1}+b_{2} X_{2}\) in terms of the real constants \(a_{1}, a_{2}, b_{1}, b_{2}\), and the parameters of the distribution.
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