Question

Suppose that the rate of return on stocks is normally dis-tributed with mean of 9% and a standard deviation of 3%. If I pick five stocks at random, what isthe probability that at least two of them will have a return of more than 12%.?

Answer 1

Answer:

The probability that at least two stocks will have a return of more than 12% is 0.1810.

Step-by-step explanation:

Let X = rate of return on stocks.

The random variable X follows a Normal distribution, N (9, 3²).

Compute the probability that a stock has rate of return more than 12% as follows:

$P(X\geq 12)=1-P(X<12)\\=1-P(\frac{X-\mu}{\sigma}<\frac{12-9}{3} )\\=1-P(Z<1)\\=1-0.8413\\=0.1587$

**Use the z table for the probability.

The probability of a stock having rate of return more than 12% is 0.1587.

Now define a random variable Y as the number of stocks that has rate of return more than 12%.

The sample size of stocks selected is, n = 5.

The random variable Y follows a Binomial distribution.

The probability of a Binomial distribution is:

$P(X=x)={n\choose x}p^{x}(1-p)^{n-x};\ x=0, 1, 2, ...$

Compute the value of P (X ≥ 2) as follows:

P (X ≥ 2) = 1 - P (X < 2)

              = 1 - P (X = 0) - P (X = 1)

              $=1-{5\choose 0}(0.1587)^{0}(1-0.1587)^{5-0}-{5\choose 1}(0.1587)^{1}(1-0.1587)^{5-1}\\=1-0.4215-0.3975\\=0.1810$

Thus, the probability that at least two stocks will have a return of more than 12% is 0.1810.