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### 45-733 PROBABILITY AND STATISTICS I Notes #4B

February 2000

1. Marginal Distributions
Definition: Continuous Marginal Distributions
f1(x) = ò-¥+¥ f(x,y)dy
f2(y) = ò-¥+¥ f(x,y)dx
Definition: Discrete Marginal Distributions
f1(xi) = åj=1,m f(xi, yj)
f2(yj) = åi=1,n f(xi, yj)

2. Example:
```
æ (1/8)x2  0 < x < 2
ç          0 < y < 3
f(x,y) = ç
ç
è   0 otherwise
```

1. Find f1(x) and f2(y)
f1(x) = ò03 (1/8)(x2)dy = (1/8)(x2)y|03 = (3/8)x2. That is:
```
æ (3/8)x2  0 < x < 2
f1(x) = ç
è   0 otherwise
```

f2(y) = ò02 (1/8)(x2)dx = [(x3)/24]|02 = 8/24 = 1/3. That is:
```
æ 1/3  0 < y < 3
f2(y) = ç
è  0 otherwise
```

Note that f2(y) is a continuous uniform distribution.