Suppose X has density function fx for a x 0 Find the density

Suppose X has density function f(x) for a=< x =<b and Y =cX + d, where c>0. Find the density function of Y.

Solution

Given that X has density function f(x) for a=< x =<b and Y = cX + d where c>0

the density of Y is denoted by f(y).

We can find density function by using cumulative distribution function.

Consider, cumulative distribution function is,

F(Y) = P(Y <= t)

= P(cX + d <= t)

= P(cX <= t - d)

= P(X <= (t-d) / c )

F(Y) = F_X [ (t - d) / c ]

differentiating with respect to t so that we get density of Y.

f_Y (t) = d / dt [ F_X (t - d) / c ]

= 1 / c f_X [ (t - d) / c ]

This is the density function of Y = cX + d.