An urn contains n red balls and m blue balls

1. An urn contains n red balls and m blue balls. They are withdrawn one at a time until a
total of r red balls have been withdrawn (here r  n). Find the probability that a total of k
balls are withdrawn. Hint: if k balls are withdrawn in total, what color is the last ball? How
many of each color are there in the other k 1 withdrawn balls?
2. Let X be a continuous random variable having cumulative distribution function F. Define
the random variable Y by Y = F(X). Show that Y is uniformly distributed over (0, 1). Hint:
what is the cumulative distribution function for Y ? You can assume that F is invertible.
Quiz 1
Form B Name
Math 130B, 6 PM
Please justify all your answers March 30, 2022
Please also write your full name on the back
1. An urn contains n red balls and m blue balls. They are withdrawn one at a time until a
total of r red balls have been withdrawn (here r  n). Find the probability that a total of k
balls are withdrawn. Hint: if k balls are withdrawn in total, what color is the last ball? How
many of each color are there in the other k 1 withdrawn balls?
2. Let X be a continuous random variable having cumulative distribution function F. Define
the random variable Y by Y = F(X). Show that Y is uniformly distributed over (0, 1). Hint:
what is the cumulative distribution function for Y ? You can assume that F is invertible.
Shiqi So
Ak =
( ? ) / ¥r ) .
r.lkit ! –
lmthK ) !
PIAK) = 17141L ) .
r.lk1) ! ( m-1hK ) !
I mtn ) !
= (7) ( Ir ) –
r
K 1m¥ )
Fyly) =P { Y Ey }
=P {7×1×14}
=P { ✗ c-FI ‘
ly) }
= Fx 11¥41 )
=
y

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