Download Chapter 6: Continuous Random Variables Suppose X is a

Chapter 6: Continuous Random Variables
Suppose X is a continuous random variable; X takes on any value in some interval of numbers. A continuous probability
distribution completely describes the random variable and is used to compute probabilities associated with the random
variable.
A probability distribution for a continuous random variable X is given by a smooth curve (density curve) or probability
density function (pfd). The curve is defined so that the probability that X takes on a value between a and b (a < b) is the
area under the curve between a and b.
1) Probability in a continuous world is area under a curve.
2) The pdf function, denoted by f (x), is not the probability that the random variable X equals the specific value x. Rather,
the function f leads to probability through area.
3) The shape of f vary considerably. The total area underneath f is 1; f (x) ≥ 0, for all x.
4) P (X = a) = 0, for any a, if X is a continuous random variable. Think of it this way: the area of a vertical line x = a
is always zero.
5) This seems like a contradiction. Certainly we can observe specific value of X, yet the probability of observing any single value is 0. Recall: probability is a limiting relative frequency. There are infinite number of values for any continuous
random variable Therefore, the limiting relative frequency of occurance of any single value is 0.
6) Therefore, P (a ≤ X ≤ b) = P (a < X ≤ b) = P (a ≤ X < b) = P (a < X < b.)
Definition: The random variable X has a uniform distribution on the interval [a, b] if f (x) =
f (x) = 0, otherwise.
1
b−a ,
for a ≤ x ≤ b, and
R
R
2
2
For uniform r.v., µ = xf (x)dx = a+b
2 , and σ = (x − µ) f (x)dx.
Cumulative distribution function=P (X ≤ x).
Example: Each watch is tested for accuracy. If the watch gains or loses time during the 24-hour testing period, it is sent
to a technician for adjustment. The time inconsistency (in seconds) is a random variable, X. The cumulative distribution
function is
1
P (X ≤ x) =
.
1 + exp−x/2
Suppose a watch is randomly selected, what is the probability that the watch is 5 seconds slow or slower? What is the
probability that the watch is more then 10 seconds fast? What is the probability that the watch is between 3 seconds slow
and 3 seconds fast?
In some problems a proability is given and we need to work backward to find a solution.
Example: Florida voters have the longest wait. Suppose the time to vote for a randomly selected person in Florida, X has
a uniform distribution between 10 an 60 minutes. Find the time t that 75% of all people have to wait at most t minutes to
vote.
Practice: Suppose X is a continuous random variable such that P (X ≤ x) = 1 − exp−x
Find P (X ≤ 2|X ≥ 1).
1
2
/5
, if x ≥ 0, and 0 otherwise.
6.2 The normal distribution
The normal distribution is the most important distribution in all of statistics.
Suppose the random variable X has mean µ and variance σ 2 , the probability density function is given by
f (x) =
2
2
1
√ exp−(x−µ) /2σ ,
σ 2π
and −∞ < x < ∞.
Suppose the random variable X has a standard normal distribution, the probability density function is given by
2
2
1
f (x) = √ exp−x /2σ ,
2π
and −∞ < x < ∞.
The steps to solve questoins involving normal random variables:
1) probability statement involving X ∼ N (µ, σ 2 ).
2) Standardize the random variable to Z ∼ N (0, 1).
3) Use cumulative probability to solve the problem.
Example: Probability calculation
a) P (Z ≤ 1.45)
b) P (Z ≥ −1.32)
c) P (−1.32 ≤ Z ≤ 1.45)
d) Find the value b such that P (Z ≤ b) = 0.80.
e)Find the 1st quartile of the standard normal distribution.
Example: Probability calculation
Suppose X ∼ N (10, 4), find P (X > 12.5), find P (9 ≤ X ≤ 10) and find value b such that P (X ≤ b) = 0.75.
Eample: suppse the SAT score for US high school student X ∼ N (1500, 100). Find P (X > 1600). Find P (1400 <
X < 1600). Find the threshold b so that P (X > b) = 0.80. The university will admit students above the threshold b so
that the upper 80% student population are above the threshold.
2