DISCUSSION 6
(March 8-9)
1 Continuous Random Variables
•Probability Density Function :
The probability density function (pdf ) f(x) describes the distribution of probability for a continu-
ous random variable. It has the properties:
1. The total area under the probability density curve is 1.
2. P(a≤X≤b) is the area under the probability density curve between aand b.
3. f(x)≥0 for all x.
•Remark : With a continuous random variable, the probability that X=xis always 0.
1. P(X=a) = 0 for all a
2. P(a≤X≤b) = P(a < X ≤b) = P(a≤X < b) = P(a < X < b)
•Median :
The median is a point with 50 percent of the distribution on either side of it. (This does not have
to be the mean.)
•100 p-th Percentile:
100 p-th percentile is a point that has area p to the left and 1-p to its right. (p is between 0 and 1)
•Quantiles are the 25th, 50th and 75 percentiles.
•The most common continuous random variable for our purposes will b e the normal random variable.
It is denoted N(µ, σ).
•To use the standard normal table to find probabilities, we almost always must standardize. We
do this by subtracting off the mean, and dividing by the standard deviation. That is:
Z=X−µ
σ
is a standard normal random variable (N(0,1)), assuming that X∼N(µ, σ).
Example 1. Determine the following probabilities for a standard normal random variable.
(a)P(0 ≤X < .5)
(b)P(.5< X < 1)
(c)P(1 < X ≤1.5)
(d)P(1.5≤X≤2)
Example 2. Determine the following for X∼N(2,3):
(a)P(0 ≤X < .5)
(b)P(1 < X ≤1.5)
(c) Find the median.
Off. Hours: R 2:30-4:30 p.m. 1 1245F MSC
