SOLUTION: The yearly inflation rate in the country follows the probability density function: f(x) = c/𝑥^3 for 𝑥

SOLUTION: The yearly inflation rate in the country follows the probability density function: f(x) = c/𝑥^3 for 𝑥 > 3 f(x) = 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Where c i

Question 1177122: The yearly inflation rate in the country follows the probability density function:
f(x) = c/𝑥^3 for 𝑥 > 3
f(x) = 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Where c is a constant.
a) Find the value of c.
b) Find an approximate probability that exactly 1 out of 5 coming years have yearly inflation rate greater than 20. (Hint: p is small)
c) Find cumulative distribution function of x.
Answer by CPhill(2189) (Show Source): You can put this solution on YOUR website!
I'll find the value of c, the probability of a specific inflation rate outcome, and the cumulative distribution function (CDF).
```python?code_reference&code_event_index=3
from sympy import integrate, Symbol, oo
# Define the variable
x = Symbol('x')
# a) Find the value of c
c = 1 / integrate(x**-3, (x, 3, oo))
print(f"a) The value of c is: {c}")
# b) Find the probability that the inflation rate is greater than 20
p = integrate(c * x**-3, (x, 20, oo))
print(f"b) The probability that the inflation rate is greater than 20 is: {p}")
# Calculate the probability that exactly one out of five years has an inflation rate greater than 20
n = 5
k = 1
prob_1_out_of_5 = (p ** k) * ((1 - p) ** (n - k)) * (5 / 1) # binomial probability formula
print(f"The probability that exactly one out of five years has an inflation rate greater than 20 is: {prob_1_out_of_5}")
# c) Find the cumulative distribution function (CDF)
cdf = integrate(c * x**-3, (x, 3, x))
print(f"c) The cumulative distribution function (CDF) is: {cdf}")
```
```text?code_stdout&code_event_index=3
a) The value of c is: 18
b) The probability that the inflation rate is greater than 20 is: 9/400
The probability that exactly one out of five years has an inflation rate greater than 20 is: 0.102711621801270
c) The cumulative distribution function (CDF) is: 1 - 9/x**2
```
Here are the results:
* **a) Value of c:**
The constant c in the probability density function is 18.
* **b) Probability of exactly one year out of five with inflation greater than 20:**
The probability that exactly one out of the next five years will have an inflation rate greater than 20 is approximately 0.1027.
* **c) Cumulative Distribution Function (CDF):**
The cumulative distribution function (CDF) for the inflation rate (x) is:
CDF(x) = 0, for x ≤ 3
CDF(x) = 1 - 9/x^2, for x > 3
Let me know if you have any further questions or need more assistance with this or another problem!
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