WW: 1Course: M339W/M389W - Intro to Financial Math Page: 1 of 1
University of Texas at Austin
Warm-up Worksheet #1
The Central Limit Theorem.
In preparation for the next class, please solve the following problems:
1.1. The Central Limit Theorem (CLT) aka the normal approximation.
Problem 1.1. A biased coin (probability of heads is 0.7) is tossed 1000 times. Write down the exact
expression for the probability that more than 750 heads have been observed. Use the normal approximation
to estimate this probability.
Solution: The random variable Xwhich equals to the number of heads is binomial with probability
p= 0.7 and n= 1000. We are interested in the probability P[X > 750]. If we split this probability among
the elementary outcomes which are >750, we get
P[X > 750] =
1000
X
i=751
P[X=i] =
1000
X
i=751 1000
i(0.7)i(0.3)1000−i.
According to the Central Limit Theorem, the random variable
X0=X−E[X]
pVar[X]=X−700
√1000 ·0.7·0.3,
is approximately normally distributed with mean 0 and standard variation 1. Therefore (note the continuity
correction),
P[X > 750] = P[X0≥750.5−700
√210 ]≈P[Z≥3.48483],
where Z∼N(0,1) is normally distributed with mean 0 and standard variation 1. Table look-up gives your
Φ(3.48) = 0.9997, so that the answer is approximately 0.0003. However, your computer will give you that
this probability is approximately equal to 0.0002.
Instructor: Milica ˇ
Cudina