Download 3.4 NORMAL DISTRIBUTION (1) A probability density function, or

3.4 NORMAL DISTRIBUTION
(1) A probability density function, or PDF is a function that represents the probability for a continuous random variable.
(2) Properties of PDF:
1. The function must be positive (or zero) everywhere and the total area between the graph of the function and the x-axis is one.
2. The probability P (a ≤ X ≤ b) that a random variable is between two values,
say a and b, is equal to the area under the graph between the boundaries a
and b.
3. For continuous random variable, P (X = a) = 0 and thus P (X ≤ a) =
P (X < a), P (X ≥ a) = P (X > a).
(3) The probability density function for a normal distribution is
(x−µ)2
1
f (x) = √ e− 2σ2
σ 2π
where µ is the mean and σ is the standard deviation.
(4) Properties of normal distribution: A normal distribution curve is a bellshaped probability density function with mean µ and standard deviation σ such
that
1. The curve is symmetric about x = µ and attains its maximum there.
2. It approaches zero on both ends, but never equals zero.
3. The standard deviation determines how “steep” or “flat” the curve is. The
curve is flatter for large σ.
(5) A standard normal distribution is a normal distribution with µ = 0 and σ = 1.We
usually denote the random variable associated with the standard normal distribution by Z.
(6) The figure below is the curve of the probability density function for the standard
normal distribution
1
2
3.4 NORMAL DISTRIBUTION
In convention, we use the following notation for the STANDARD normal distribution
A(b) = P (Z ≤ b).
Then A(0) = 0.5, P (Z ≥ b) = 1 − A(b), P (a ≤ X ≤ b) = A(b) − A(a).
Generally, if X is normally distributed with mean µ and standard deviation σ,
then the variable (X − µ)/σ is standard normally distributed. It follows that
a − µ
b − µ
P (a ≤ X ≤ b) = P
≤X≤
σ
σ
a − µ
b − µ
−A
=A
σ
σ
(7) To find P (a ≤ Z ≤ b) (standard normal distribution)
1. Type 2ND and then VARS to get to the distribution menu.
2. Select option 2, normcdf..
3. Type a, comma, b, Close the parentheses and hit Enter.
To enter ±∞, we usually enter 1E99 or -1E99.To enter “-”, type (-). To enter
“E”, type 2ND, and then the comma “,”.
(8) In general, to find P (a ≤ X ≤ b) where X is a normally distributed random
variable with mean µ and standard deviation σ
1. Type 2ND and then VARS to get to the distribution menu.
2. Select option 2 or scroll down to normcdf and hit ENTER.
3. Type a, comma, b, comma, µ, comma, σ.
4. Close the parentheses and hit ENTER.
(9) Conversely, we can get the value of a for given P (X ≤ a). In other words, if we
know P (X ≤ a) = p but a is unknown, then we can find the corresponding a.
Suppose X is a random variable with normal distribution with mean µ and
standard deviation σ. Find a such that P (X ≤ a) = p
1. Type 2ND and then VARS to get to the distribution menu.
2. Selection option 3 or scroll down to invNorm and hit ENTER.
3. Type p, a comma, µ, a comma, and σ.
4. Close the parentheses and hit ENTER.
Guchao Zeng; Department of Mathematics, Texas A&M University, College Station, TX
77843, USA; [email protected]