Question

The price of a stock is uniformly distributed between $30 and $40. a. Write the probability density function, f(x), for the price of the stock. b. Determine the expected price of the stock. c. Determine the standard deviation for the stock. d. What is the probability that the stock price will be between $34 and $38

Answer 1

a)
`f(x) = (1)/(40-30), 30 \leq x \leq 40`

b)
`E(X) = (a+b)/(2)= (30+40)/(2)=35`

c)
`Var(X) = ((b-a)^2)/(12)= ((40-30)^2)/(12)= 8.33`

And the deviation would be:

`Sd(X) = √(8.33)= 2.887`

d)
`P(34< X <38) = F(38) -F(34)= (38-30)/(10) -(34-30)/(10)= 0.8-0.4=0.4`

Explanation:

For this case we define the random variable X with this distribution:

`X \sim Unif (a=30, b=40)`

Part a

The density function since is an uniform distribution is given by:

`f(x) = (1)/(40-30), 30 \leq x \leq 40`

Part b

The expected value is given by:

`E(X) = (a+b)/(2)= (30+40)/(2)=35`

Part c

The variance is given by:

`Var(X) = ((b-a)^2)/(12)= ((40-30)^2)/(12)= 8.33`

And the deviation would be:

`Sd(X) = √(8.33)= 2.887`

Part d

For this case we want this probability:

`P(34< X <38)`

And we can use the cumulative distribution function given by:

`F(x)= (x-30)/(40-30), 30 \leq X \leq 40`

And using this we got:

`P(34< X <38) = F(38) -F(34)= (38-30)/(10) -(34-30)/(10)= 0.8-0.4=0.4`