Question

Let X be a standard normal random variable so that X N(0; 1). For this problem you may
want to refer to the table provided on Canvas. Recall, that (x) denotes the standard normal
CDF.
(a) Find (1:45).
(b) Find x, such that (x) = 0:4.
(c) Based on the fact that (1:645) = 0:95 nd an interval in which X will fall with 95%
probability.
(d) Find another interval (dierent from the one in (c)) into which X will fall with 95% prob-
ability. Hint you may want to consider the symmetric properties of the standard normal
distribution.
(e) Find an interval with lower bound -1.9 into which X will fall with 95% probability.
(f) Find an interval with upper bound 2.1 into which X will fall with 95% probability.
(g) Among all possible intervals into which X falls with 95% probability, nd the shortest one.

Answer 1

Answer:

Let X be a standard normal random variable so that X N(0; 1).

For this problem you may want to refer to the table provided on Canvas. Recall, that (x) denotes the standard normal CDF.

Referring z - score table:

(a).

(b).

(c).

Since,

Interval where 95% of values falls is

(d).

There are infinite number of intervals where X falls with probability is 0.95

One of them is

(-1.96, 1.96)

which happens to be the smallest interval.

(e).

Let (-1.9,k₁) be the interval, with lower bound -1.9, in which X will fall with 95% probability.

So, we have

P(-1.9<X<k₁)=0.95

(=cdf of z)

(Since, )

(From normal table values)

So, the required interval is =(-1.9,2.028)

(f).

Let (k₂,2.1) be the interval, with upper bound 2.1, in which X will fall with 95% probability.

So, we have

P(k₂<X<2.1)=0.95

So, the required interval is =(-1.851,2.1)