Question

Suppose X has a standard normal distribution. The pnorm(), norm(), and qnorm() functions should be useful in the following. Remember, if X has a standard normal distribution then E(X) = 0 and Var(x)=1. a) What is the expected value of Xt1 ? b) What is the variance of X c) What is the expected value of xo? d) Using R, calculate the probability X > .83) e) Using R, calculate the probability that -0.2 <x< 1.1 f) Using R, calculate the probability that 0 < g) If the probability that X > t = .29 then what is t? h) Copy your R script for the above into the text box here.

Answer 1

a) E((X+1)/3) = E(X)/3 + 1/3 = 0+1/3 = 1/3=0.3333

b) V((X+1)/3) = V(X/3) = 1*(1/3)^2 = 1/9=0.1111

c) E(X^6) = 15

d) P(X>0.83) = 0.2033

e) P(-0.2<X<1.1) = 0.4436

f) P(0<(X+1)/3<1) = P(-1<X<2) = 0.8186

g) P(X>t)= 0.29 implies t= 0.5534

h) The R codes are as follows -

# a) and b)
E_x=0
V_x=1
y=(x/3) +(1/3)
E_y= (0/3) + (1/3)
V_y = (1/3)^2
E_y
V_y

#c)
x=rnorm(1000000,0,1)
y=x^6
mean(x)
mean(y)
#The mean of X^6 goes close to 15

#d)
w=pnorm ( 0.83,lower.tail=FALSE)
w

#e)
r1=pnorm ( 1.1,lower.tail=TRUE)
r2=pnorm ( -0.2,lower.tail=TRUE)
r1-r2

#f)
s1=pnorm ( 2,lower.tail=TRUE)
s2=pnorm ( -1,lower.tail=TRUE)
s1-s2

#g)
m=qnorm(0.29,lower.tail=FALSE)
m