Let's break this down part by part.

(a) Identifying the Graph of the Uniform Density Function

For a uniform probability distribution over the interval $[0, 1]$, the probability density function (PDF) is a rectangle. The area under the graph must equal 1 (since the total probability must be 1).

For a uniform distribution on $[0, 1]$, the height of the rectangle is determined by the formula for the height of the uniform distribution:

$\text{Height} = \frac{1}{\text{Range of x}} = \frac{1}{1 - 0} = 1.$

Thus, the graph is a rectangle that spans from $x = 0$ to $x = 1$, and the height of the rectangle is 1. The correct graph should reflect this.

Let's analyze the choices:

• Option A: The rectangle only goes from $x = 0$ to $x = 0.4$, which is not a valid uniform distribution over $[0, 1]$.
• Option B: The rectangle spans from $x = 0$ to $x = 1$, with a height of 0.8, which is incorrect because the height should be 1, not 0.8.
• Option C: This seems to represent a uniform distribution over $[0, 1]$ with the height equal to 1, making it the correct graph.

Thus, the correct answer is Option C.

(b) Probability of Generating a Number Between 0.44 and 0.58

For a uniform distribution over $[0, 1]$, the probability of generating a number between any two values is simply the length of the interval. The formula for the probability $P(a \leq X \leq b)$ for a uniform distribution is:

$P(a \leq X \leq b) = b - a.$

Here, we are asked to find the probability of generating a number between 0.44 and 0.58. So:

$P(0.44 \leq X \leq 0.58) = 0.58 - 0.44 = 0.14.$

Thus, the probability of generating a number between 0.44 and 0.58 is 0.14.

(c) Probability of Generating a Number Greater Than 0.91

For the uniform distribution over $[0, 1]$, the probability of generating a number greater than a certain value $x$ is the length of the interval $[x, 1]$. The formula for this probability is:

$P(X > x) = 1 - x.$

For $x = 0.91$, we get:

$P(X > 0.91) = 1 - 0.91 = 0.09.$

Thus, the probability of generating a number greater than 0.91 is 0.09.

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Final Answers:

(a) Option C is the correct graph for the uniform distribution.
(b) The probability of generating a number between 0.44 and 0.58 is 0.14.
(c) The probability of generating a number greater than 0.91 is 0.09.

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Would you like more details or clarification on any of the steps? Here are some related questions for further exploration:

1. How would the graph change if the uniform distribution was over a different interval, say $[a, b]$?
2. What happens to the height of the uniform distribution if the range of $X$ is changed to $[0, 5]$?
3. How do you calculate the cumulative distribution function (CDF) for a uniform distribution?
4. What are some practical examples of uniform distributions in real life?
5. How does the uniform distribution compare to other distributions like the normal or exponential distributions?

Tip: When working with uniform distributions, remember that the probability is proportional to the length of the interval being considered!