#Exercícios de "Noções de Probabilidade e Estatística" # #Seção 6.2 #Exercicio 1 # X ~ U[0,4] ########################################################################### # a) P(X>2) = P(2 < X <4) x <- seq(-2.5,4.5,.01) y <- cbind(dunif(x,0,4)) # corresponde os valores de x com y. plot(x,y,type="l",col=1,main="Exercício 1.a",sub= "X ~ Uniforme[0,4]") text(-2.4,.25,paste("P(X>2) = P(2 < X < 4)"),adj=0) text(-2.4,.23,expression(paste("P(X>2)") == integral(frac(1,4)*dx,2,4)),adj=0) text(-2.4,.21,paste("P(X>2) = 0,5"),adj=0) ########################################################################### # b) P(X >= 2) = P(X > 2) = P(2 < X <4) x <- seq(-2.5,4.5,.01) y <- cbind(dunif(x,0,4)) # corresponde os valores de x com y. plot(x,y,type="l",col=1,main="Exercício 1.b",sub= "X ~ Uniforme[0,4]") text(-2.4,.25,expression(P*(X >= 2) == P *(X > 2)),adj=0) text(-2.4,.23,expression(P*(X >= 2) == paste("0,5")),adj=0) ########################################################################### # c) P(1 < X < 2), temos que encontrar a área desta função neste segmento. x <- seq(-2.5,4.5,.01) y <- cbind(dunif(x,0,4)) # corresponde os valores de x com y. plot(x,y,type="l",col=1,main="Exercício 1.c",sub= "X ~ Uniforme[0,4]") text(-2.4,.24,expression(paste("P(1 < X < 2)") == integral(frac(1,4)*dx,1,2)),adj=0) text(-2.4,.22,paste("P(1 < X < 2) = 0,25"),adj=0) ########################################################################### # d) P(1 < X < 2| X < 3) x <- seq(-2.5,4.5,.01) y <- cbind(dunif(x,0,4)) # corresponde os valores de x com y. plot(x,y,type="l",col=1,main="Exercício 1.c",sub= "X ~ Uniforme[0,4]") text(-2.4,.24,paste("P(1 < X < 2| X < 3)"),adj=0) text(-2.4,.22,expression(paste("= ", frac(paste("P(1 < X < 2)"),paste("P(0 < X < 3)")))),adj=0) a <- punif(2,1,5) b <- punif(3,0,4) text(-2.4,.19,expression(paste("P(1 < X < 2| X < 3)") == frac(paste("0,25"),paste("0,75"))),adj=0) text(-2.4,.165,paste("P(1 < X < 2| X < 3) = 0,333"),adj=0) ########################################################################### # d) P(X < 3|1 < X < 2) x <- seq(-2.5,4.5,.01) y <- cbind(dunif(x,0,4)) # corresponde os valores de x com y. plot(x,y,type="l",col=1,main="Exercício 1.d",sub= "X ~ Uniforme[0,4]") text(-2.4,.24,paste("P(X < 3|1 < X < 2)"),adj=0) text(-2.4,.22,expression(paste("= ", frac(paste("P(1 < X < 2)"),paste("P(1 < X < 2)")))),adj=0) text(-2.4,.20,paste("P(X < 3|1 < X < 2) = 1"),adj=0) ########################################################################### ########################################################################### # #Exercício 2 ########################################################################### # a) X ~ U[0,10] # primeiros 500 metros -> P(X < 0,5) = P(0 < x < 0,5) # 3 Km centrais -> min(x)=((10-3)/2), max(x) = ((10-3)/2)+3. # P(3.5 < X < 6.5), ambas probabilidades são conseguidas pela obtenção da área # da função entre os intervalos de cada função. x <- seq(-6,10.5,.01) y <- cbind(dunif(x,0,10)) # corresponde os valores de x com y. plot(x,y,type="l",col=1,main="Exercício 2.a",sub= "X ~ Uniforme[0,10]") text(-5.9,.095,paste("Primeiros 500 metros"),adj=0) text(-5.9,.090,paste("P(X < 0,5) = P(0 < x < 0,5)"),adj=0) punif(.5,0,10) text(-5.9,.085,paste("P(X < 0,5) = 0,05"),adj=0) a <- punif(6.5,0,10) - punif(3.5,0,10) text(-5.9,.070,paste("3 quilômetros centrais"),adj=0) text(-5.9,.065,paste("P(3.5 < X < 6.5)","=",a),adj=0) ########################################################################### ########################################################################### # # Exerccício 3 # # X ~ U[5,15] ########################################################################### # a) P(X < 10) = P (5 < X <10) x <- seq (-2,16,.01) y <- cbind(dunif(x,5,15)) plot(x,y,type="l",col=1,main="Exercício 3.a",sub="X ~ U[5,15]") text(-2,.1,paste("P(X < 10) = P (5 < X <10)"),adj=0) text(-2,.095,paste("P(X < 10) = ", punif(10,5,15)),adj=0) ########################################################################### # b) P(X > 12) = P (12 < X <15) plot(x,y,type="l",col=1,main="Exercício 3.b",sub="X ~ U[5,15]") text(-2,.1,paste("P(X > 12) = P (12 < X <15)"),adj=0) text(-2,.095,paste("P(X > 12) = ", punif(15,5,15)-punif(12,5,15)),adj=0) ########################################################################### # c) P(X > 7| X<10) = P (7 < X <10)/ P(5 < X < 10) plot(x,y,type="l",col=1,main="Exercício 3.c",sub="X ~ U[5,15]") text(-2,.1,paste("P(X > 7| X<10)"),adj=0) text(-2,.090,expression(paste("=",frac(paste("P(7 < X <15)"), paste("P(5 < X < 10)")))),adj=0) text(-2,.080,paste("P(X > 7| X<10) = ", (punif(10,5,15)-punif(7,5,15))/(punif(10,5,15))),adj=0) ########################################################################### ########################################################################### # # Exerccício 4 #