Problem 2E

Let X have the pdf f (x) = xe−x2/2, 0 < x < ∞. Find the pdf of Y = X2.

ST 701 Week Five Notes MaLyn Lawhorn September 12, 2017 and September 14, 2017 Last Time ▯ Cumulative Distribution Function (CDF) – denoted X (x) – gives the probability that X ▯ x, x 2 R – Properties: 1. non-decreasing function 2. F(▯1) = 0, F(+1) = 1 3. right-continuousX F (x) = ▯#0X(x + ▯) 4. Consider P(a < X ▯ b) ▯ We know that (▯1;b] = (▯1;a] [ [a;b]. ▯ So, FX(b) = X (a) [ [a;b], and thus P(a < X ▯ b)X= F (b)X▯ F (a) 5. P(X = x) = F (x)▯F (x▯) where F (x▯) is the limit lim F (x▯▯). This X X X ▯#0 X can also be described as the gap betwe