Given the mathematical function f(x) = x, determine the upper limit for x to make this function a probability density function starting from x = 0 (i.e., determine b given a = 0).

b) Properly state the probability density function over all values of x 0.

c) Graph the probability density function over all values of x 0.

d) Using the above probability density function determine the probability that x will be between 0 and 1.

e) Using the above probability density function determine the probability that x will be between 1 and 4.

Find the mean, variance, and standard deviation for the probability density function , f(x)=0.1,0X10.Graph the probability density function, then find and graph the cumulative distribution function on the same axes.

2. Find the mean, variance, and standard deviation for the probability density function f(x)=(1/2)e^(x/2), 0X Graph the probability density function, then find and graph the cumulative distribution function on the same axes.

3. Find the mean, variance, and standard deviation for the probability density function. ,f(x)=1.5x^2, 1X1 Graph the probability density function, then find and graph the cumulative distribution function on the same axes.

Determine whether f(x) = ] 12/ (x (4+x ) ) x20 is a probability density function and justify your answer. 0 X