The chi-square(χ2) distribution with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables.
If Zk are independent and standard normal random variables, the sum of their squares is distributed according to the chi-square distribution with k degrees of freedom.
The following graphs are some chi-square distributions for various degrees of freedom.
The following graphs are some chi-square distributions for various degrees of freedom.
What is the probability for random variable χ² < 4 in a χ²-distribution with degree of freedom of 5?
The following will be the input:
The following will be the result of Statistics Study:
CHI-SQUARE DISTRIBUTION
Your Input:
χ² = 4
sig. level =
degree of freedom: k = 5
Probability Density Function (PDF)
x^(k/2-1)·e^(-x/2)
f(x;k) = ------------------
2^(k/2)·Γ(k/2)
where Γ() is the gamma function.
x f(x) = height of the graph
------------+-------------------+
0 0.0000 |
1 0.0807 |**********
2 0.1384 |******************
3 0.1542 |********************
4 0.1440 |*******************
5 0.1220 |****************
6 0.0973 |*************
7 0.0744 |**********
8 0.0551 |*******
9 0.0399 |*****
10 0.0283 |****
11 0.0198 |***
12 0.0137 |**
13 0.0094 |*
14 0.0064 |*
15 0.0043 |*
16 0.0029 |
17 0.0019 |
18 0.0013 |
19 0.0008 |
20 0.0005 |
------------+-------------------+
f(4, 5) = 0.143976
Cumulative Distribution Function (CDF)
1
F(x;k) = --------·γ(k/2, x/2)
Γ(k/2)
where γ() is the incomplete gamma function.
x F(x) = P(< x)
------------+-------------------+
0 0.0000 |
1 0.0374 |*
2 0.1509 |***
3 0.3000 |******
4 0.4506 |*********
5 0.5841 |************
6 0.6938 |**************
7 0.7794 |****************
8 0.8438 |*****************
9 0.8909 |******************
10 0.9248 |******************
11 0.9486 |*******************
12 0.9652 |*******************
13 0.9766 |********************
14 0.9844 |********************
15 0.9896 |********************
16 0.9932 |********************
17 0.9955 |********************
18 0.9971 |********************
19 0.9981 |********************
20 0.9988 |********************
------------+-------------------+
| ___
| -- ---
| - ----
| - | ----
|- | ---_____
+-------------+--------------
χ²
χ² = 4
Degree of Freedom = 5
P(χ² < 4) = 0.450584
P(χ² > 4) = 0.549416
Properties of the PDF
Mean = k = 5
Median ≈ k(1-2/9k)² = 4.3625
Mode = max(k-2, 0) = 3
Variance = 2k = 10
Skewness = √(8/k) = 1.2649
Ex. kurtosis = 12/k = 2.4