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$X\sim B(n,p)?$

(b) Find$E\left(X\right){\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\sigma \left(X\right)$ .

(c) Let Y be the random variable which models the change in share price. Then$\u04ae-0.2X-0.2(n-X)$ because 0.2X is the total increase in share price and $0.2(n-X)$ is the total decrease in share price. Simplify the expression for Y in terms of X. Then using (b), find E(Y) and $\sigma \left(Y\right)$ .

(a) What is n and p if

(b) Find

(c) Let Y be the random variable which models the change in share price. Then

Natalie Yamamoto

Beginner2021-12-15Added 22 answers

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,$X\sim B(5,0.6)$

Part b:

We know the mean and variance of binomial distribution are given by,

$\text{Mean}=E\left(X\right)=np=5\times 0.6=3$ (1)

Variance$=V\left(X\right)=npq=5\times 0.6\times 0.4=1.2$ (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,

$Y=0.2X-0.2(n-X)$

$=0.2X-0.2n+0.2X$

$=0.4X-0.2\left(5\right)$

$=0.4X-1.0$

Thus, we have mean and variance of random variable Y as,

Mean$\left(Y\right)=E\left(Y\right)=E(0.4X-1.0)$

$=0.4E\left(X\right)-1.0$

$=0.4\times 3-1.0$ (From (1))

$=0.2$

Variance$\left(Y\right)=V\left(Y\right)=V(0.4X-1.0)$

$=0.42V\left(X\right)$

$=0.16\times 1.2$ (Form (2))

$=0.192$

Thus,$\sigma \left(Y\right)=\sqrt{V\left(Y\right)}=\sqrt{0.192}=0.4382$

Part a:

Let the random variable X denote the number of increase in the price of the share over the 5 months.

Thus,

Thus,

Part b:

We know the mean and variance of binomial distribution are given by,

Variance

Thus,

Step 2

The random variable y is defined as,

Thus, we have mean and variance of random variable Y as,

Mean

Variance

Thus,

soanooooo40

Beginner2021-12-16Added 35 answers

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