Question

Assume that the weight loss for the first month of a diet program varies between 6 pounds and 14 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of pounds lost.

1) P(x = 7)

2) Between 8.25 po9unds and 12 pounds.

3) Greater than 10.50 pounds.

Please show work.

Answer 1

Hello!

For this uniform distribution, the shape of the density function is a line. We can begin by identifying key elements of the distribution:

Range: 6 to 14 pounds

Formula of density function:

y = 1/Range

1 - (14 - 6) = 1/8 = 0.125

f(x) = 0.125

1)

P(x = 7)

A uniform distribution is a continuous distribution, so the probability of x being an EXACT value is approximately 0.

This is the same as if we took the following integral. (Finding the area under a curve with the same start and end point)

`\int\limits^a_a {f(x)} \, dx = 0`

Therefore, P(x = 7) = 0.

2)
We can think of this as finding the area of a rectangle.

The width would be the difference between 8.25 pounds and 12 pounds:
W = 12 - 8.25 = 3.75

The height would be the function (y = .125), or:
h = .125

Using the equation for the area of a rectangle: (A = h * w)

A = 3.75 * .125 = 0.469

Thus, P(8.25 < x < 12) = 0.469

3)
To find the probability GREATER than 10.50 pounds, we can subtract this value from the max value for the width:
W = 14 - 10.50 = 3.50

The height is the same as above:
h = .125

Solve for the area:
A = 3.50 * .125 = 0.4375

P(x > 10.5) = 0.4375