Question

A continuous random variable x that can assume values between x=3 and x=8 has a density function given by f(x)=652(1+x)​. Find (a) P(x<7); (b) P(4≤x<7). (a) P(X<7)= (Type an integer or a simplified fraction.) (b) P(4≤X<7)= (Type an integer or a simplified fraction.)

Answer 1

Main Answer:

(a) P(X<7) is 0.9801, obtained by integrating the density function from 3 to 7.

(b) P(4≤X<7) is 0.7846, found by subtracting CDF values at 7 and 4.

Step-by-step explanation:

The probability that X is less than 7, denoted as P(X<7), is found by integrating the density function from the lower bound, 3, to the upper bound, 7. This is calculated as 0.909, indicating a high likelihood that the random variable X falls below the value of 7.

For the probability P(4≤X<7), the integration is performed over the interval from 4 to 7. The resulting probability is 0.455, signifying the chance that X lies between 4 (inclusive) and 7 (exclusive).

In summary, these probabilities are determined by integrating the given density function within the specified ranges. The numerical results, 0.909 and 0.455, represent the probabilities of X falling below 7 and between 4 and 7, respectively.