Question

Please help me solve this question thanks

Problem 3. The random variable X has density function f given by 0, elsewhere (a) Assuming that θ 0.8, determine K (b) Find Fx(t), the c.d.f. of X (c) Calculate P(0.4SX 0.8)

Answer 1

a)

for above to be valid: $\int_{0}^{1}$ f(y) dy =1

$\int_{0}^{1}$ f(y) dy =$\int_{0}^{0.8}$0.8y² dy +$\int_{0.8}^{1}$ Ky² dy =0.8y³/3 |^0.8₀ +Ky³/3 |¹_0.8 =0.8⁴/3 +K(1/3-0.8³/3) =1

K=5.3082

b)

for 0 < y< 0.8

F(y) =$\int_{0}^{y}$0.8y² dy =0.8y³/3 |^y₀ =0.8y³/3

for 0.8 <y<1

F(Y) =$\int_{0}^{0.8}$0.8y² dy +$\int_{0.8}^{y}$ 5.3082 y² dy = 0.8y³/3 |^0.8₀ +5.3082*y³/3 |^y_0.8 =5.3082*y³/3-0.7694

c)

P(0.4<Y<0.8)=F(0.8)-F(0.4)=0.8*0.8³/3 -0.8*(0.4)³/3 =0.1195