is there anyone who can give some clues on this problem Let

is there anyone who can give some clues on this problem?

Let X1, X2, ?, Xk be random variables of the continuous type, and let f1(x), f2(x), ?, fk(x) be their corresponding pdfs, each with sample space S = (-infinity, infinity). Also, let c1, c2, ?, ck be nonnegative constants such that sigma k I = 1 ci = 1. (a) Show that Sigma k i=1 cifi(x) is a pdf of a continuous-type random variable on S. (b) If X is a continuous-type random variable with pdf Sigma k i=1 cifi(x) on S, E(Xi) = Mui and Var(Xi) = sigma^2 I for I = 1, .., k, find the mean and the variance of X. is there anyone who can give some clues on this problem?

Solution

a) X1, x2.... xn are random variables with pdf f1(x),fz(x),...fn(x)

Each xi is continuous

Hence c1x1+c2x2+c3x3+....+cnxn, a linear combination of x1, x2,...xn with non negative scalars c1, c2.. cn form a

pdf of again a continuous random variable on S.

b) E(X) = E(c1x1+c2x2+c3x3+...+cnxn)

=E( )

= i

Var(x) =