gorovogpg

2021-12-14

Consider a share that is modelled by a binomial random variable. The probability that the share increases in value by 20¢ in one month is 0.6. The probability that it decreases in value by 20¢ in one month is 0.4. The share is held for 5 months then sold. Let X denote the number of increases in the price of the share over the 5 months. В (п,
(a) What is n and p if $X\sim B\left(n,p\right)?$
(b) Find $E\left(X\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\sigma \left(X\right)$.
(c) Let Y be the random variable which models the change in share price. Then $Ү-0.2X-0.2\left(n-X\right)$ because 0.2X is the total increase in share price and $0.2\left(n-X\right)$ is the total decrease in share price. Simplify the expression for Y in terms of X. Then using (b), find E(Y) and $\sigma \left(Y\right)$.

Natalie Yamamoto

Step 1
Part a:
Let the random variable X denote the number of increase in the price of the share over the 5 months.
Thus, $n=\text{total number of months}=5$
$p=\text{the probability that the share increases in value by 20 in one month}$
$=0.6$
$q=\text{the probability that the share decreases by value 20 in one month}$
$=0.4$
Thus, $X\sim B\left(5,0.6\right)$
Part b:
We know the mean and variance of binomial distribution are given by,
$\text{Mean}=E\left(X\right)=np=5×0.6=3$ (1)
Variance $=V\left(X\right)=npq=5×0.6×0.4=1.2$ (2)
Thus,
$\sigma \left(X\right)=\sqrt{V\left(X\right)}=\sqrt{1.2}=1.0954$
Step 2
The random variable y is defined as,
$Y=0.2X-0.2\left(n-X\right)$
$=0.2X-0.2n+0.2X$
$=0.4X-0.2\left(5\right)$
$=0.4X-1.0$
Thus, we have mean and variance of random variable Y as,
Mean $\left(Y\right)=E\left(Y\right)=E\left(0.4X-1.0\right)$
$=0.4E\left(X\right)-1.0$
$=0.4×3-1.0$ (From (1))
$=0.2$
Variance $\left(Y\right)=V\left(Y\right)=V\left(0.4X-1.0\right)$
$=0.42V\left(X\right)$
$=0.16×1.2$ (Form (2))
$=0.192$
Thus, $\sigma \left(Y\right)=\sqrt{V\left(Y\right)}=\sqrt{0.192}=0.4382$

soanooooo40

The price-to-earnings ratio ($\frac{P}{E}$ ratio) of a stock is given by

where denotes the price per share of the stock and denotes the earnings per share. Estimate the change in the $\frac{P}{E}$ ratio of a stock if its price increases from $60 share to$62 share while its earnings decrease from $4 share to$3.80 share.

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