5.2 The uniform distribution  (Page 3/16)

Chapter review

If X has a uniform distribution where a < x < b or a ≤ x ≤ b , then X takes on values between a and b (may include a and b ). All values x are equally likely. We write X ∼ U ( a , b ). The mean of X is $\mu =\frac{a+b}{2}$ . The standard deviation of X is $\sigma =\sqrt{\frac{{(b-a)}^2}{12}}$ . The probability density function of X is $f(x)=\frac{1}{b-a}$ for a ≤ x ≤ b . The cumulative distribution function of X is P ( X ≤ x ) = $\frac{x-a}{b-a}$ . X is continuous.

The probability P ( c < X < d ) may be found by computing the area under f ( x ), between c and d . Since the corresponding area is a rectangle, the area may be found simply by multiplying the width and the height.

Formula review

X = a real number between a and b (in some instances, X can take on the values a and b ). a = smallest X ; b = largest X

X ~ U (a, b)

The mean is $\mu =\frac{a+b}{2}$

The standard deviation is $\sigma =\sqrt{\frac{{(b\text{ – }a)}^2}{12}}$

Probability density function: $f(x)=\frac{1}{b-a}$ for $a\le X\le b$

Area to the Left of x : P ( X < x ) = ( x – a ) $\left(\frac{1}{b-a}\right)$

Area to the Right of x : P ( X > x ) = ( b – x ) $\left(\frac{1}{b-a}\right)$

Area Between c and d : P ( c < x < d ) = (base)(height) = ( d – c ) $\left(\frac{1}{b-a}\right)$

Uniform: X ~ U ( a , b ) where a < x < b

• pdf: $f\left(x\right)=\frac{1}{b-a}$ for a ≤ x ≤ b
• cdf: P ( X ≤ x ) = $\frac{x-a}{b-a}$
• mean µ = $\frac{a+b}{2}$
• standard deviation σ $=\sqrt{\frac{{(b-a)}^2}{12}}$
• P ( c < X < d ) = ( d – c ) $(\frac{1}{b–a})$

References

McDougall, John A. The McDougall Program for Maximum Weight Loss. Plume, 1995.

Use the following information to answer the next ten questions. The data that follow are the square footage (in 1,000 feet squared) of 28 homes.

The sample mean = 2.50 and the sample standard deviation = 0.8302.

The distribution can be written as X ~ U (1.5, 4.5).

Use the following information to answer the next eight exercises. A distribution is given as X ~ U (0, 12).

Draw the graph of the distribution for P ( x >9).

Got questions? Get instant answers now!

Use the following information to answer the next eleven exercises. The age of cars in the staff parking lot of a suburban college is uniformly distributed from six months (0.5 years) to 9.5 years.