Show that if X1, X2, …, Xp are independent random variables and Y=c1 X1+c2 X2+⋯+cp Xp, V(Y)=c1^2

step 1 Hi guys, in this question we are given back x1 x2 up to xp are independent random variables and

Show that if X1, X2, …, Xp are independent random variables...

Show that if $X_{1}, X_{2}, \ldots, X_{p}$ are independent random variables and $Y=c_{1} X_{1}+c_{2} X_{2}+\cdots+c_{p} X_{p}$, $$ V(Y)=c_{1}^{2} V\left(X_{1}\right)+c_{2}^{2} V\left(X_{2}\right)+\cdots+c_{p}^{2} V\left(X_{p}\right) $$ You may assume that the random variables are continuous.

Show that if $X_{1}, X_{2}, \ldots, X_{p}$ are independent random variables and $Y=c_{1} X_{1}+c_{2} X_{2}+\cdots+c_{p} X_{p}$,
$$
V(Y)=c_{1}^{2} V\left(X_{1}\right)+c_{2}^{2} V\left(X_{2}\right)+\cdots+c_{p}^{2} V\left(X_{p}\right)
$$
You may assume that the random variables are continuous.

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Key Concepts

Variance

Variance measures the spread or dispersion of a random variable around its mean. It is defined as the expected value of the squared deviation from the mean, and for any constant multiplied by a random variable, the variance scales by the square of that constant. This property is crucial when computing the variance of linear combinations of random variables.

Independent Random Variables

When random variables are independent, the covariance between any two distinct variables is zero. This elimination of cross-product terms simplifies the computation of the variance of their sum, as it allows the variance to be expressed solely as the sum of the individual variances scaled by the square of their respective coefficients.

Linear Combination of Random Variables

A linear combination involves summing random variables each multiplied by a constant coefficient. The calculation of the variance of a linear combination utilizes the property that the variance of a scaled random variable is the square of the coefficient times the variance of the variable, combined with the independence property to eliminate cross terms, leading to a straightforward sum of variances.

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