6. Normal Distribution and Continuous Random Variables

Uniform Distribution

6. Normal Distribution and Continuous Random Variables

Uniform Distribution

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Multiple Choice

Shade the area corresponding to the probability listed, then find the probability.
$P of open paren X is less than 7.5 close paren$

Graph showing a shaded area under a uniform distribution curve, indicating P(X<7.5) with axes labeled for probability density and x. ; $P of open paren X is less than 7.5 close paren equals 0.15$

Graph showing a uniform distribution with shaded area representing P(X<7.5) under the probability density curve. ; $P of open paren X is less than 7.5 close paren equals 0.25$

Graph showing a uniform distribution with shaded area representing P(X<7.5) under the probability density curve. ; $P of open paren X is less than 7.5 close paren equals 0.5625$

Graph showing a shaded area under a uniform distribution curve, indicating P(X<7.5) with axes labeled for probability density and x. ; $P of open paren X is less than 7.5 close paren equals 0.5625$

Verified step by step guidance

Step 1: Identify the type of distribution represented in the graph. The graph shows a triangular probability density function (PDF), which is a continuous distribution. The area under the curve represents probabilities.

Step 2: Recognize that the problem asks for the probability P(X < 7.5). This corresponds to the shaded area under the curve from the start of the distribution (x = 2.5) to x = 7.5.

Step 3: Break the shaded area into geometric shapes for calculation. The shaded region forms a triangle. The base of the triangle is from x = 2.5 to x = 7.5, and the height is determined by the value of the PDF at x = 7.5.

Step 4: Use the formula for the area of a triangle: Area = (1/2) × base × height. The base is (7.5 - 2.5 = 5), and the height is the value of the PDF at x = 7.5, which is 0.15.

Step 5: Calculate the area using the formula. This area represents the probability P(X < 7.5). Substitute the values into the formula to find the probability.