7.3 Probability Distributions for
Continuous Random Variables
Thursday, July 13, 2017
Recall
• Continuous random variable has as its set of
possible values an entire interval.
• Ex: weight of a newborn, x (in pounds)
-suppose only recorded to nearest pound
-then x’s are only whole numbers
-prob. dist. can be pictured as prob.
histogram where area of each rectangle
is probability of corresponding weight
-total area of all rectangles is 1.
-probability that a weight is between two
values, like 6 and 8, is the sum of the
corresponding rectangular areas.
• (fig, 7.5a pg. 362)
• Now suppose the weight is measured to
nearest tenth of a pound
– There are many more possible weight values
than before, the rectangles of the probability
histogram are much narrower and the histogram
looks much smoother
– The histogram can also be drawn where area of
each rectangle equals the corresponding
probability and total area of all rectangles = 1
• Figure 7.5c shows what happens as weight is
measured to a greater and greater degree of
accuracy
• The sequence of prob. histogram approaches
a smooth curve.
• Probability distribution for a
continuous r.v. is specified by a
mathematical function f(x) and called the
density function. The graph is a smooth
curve called the density curve.
Density function/Curve
• Probability of a range of values for x is
found by determining area under the
density function over that interval
Requirements, which must be met,
for a density curve:
bg
1. f x  0 (curve can’t dip below horizontal
axis)
2. Total area under curve = 1
The probability that x falls in any interval is
the area under the density curve and
above the interval.
3 Events for Probability distribution
for of a continuous r.v.
1. The event the random variable is
between 2 numbers
a<x<b
2. The event the r.v. is less than a number
x<a
3. The event the r.v. is greater than a
number
x>b
• Ex. 7.7 Application Processing Times pg. 363
uniform dist.
b g
b g
P x  3 means same as P x  3
– “or equal to” makes no difference with
continuous r.v. (no area at 3, it takes an
“interval” to get area)
• Cumulative area – bottom pg. 364
P a  x  b =cumulative area to left of b –
cumulative area to left of a
b
g