# 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 They are withdrawn one at a time until a

total of r red balls have been withdrawn (here r n).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balls are withdrawn. Hint: if k balls are withdrawn in total, what color is the last ball?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many of each color are there in the other k 1 withdrawn balls?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 They are withdrawn one at a time until a

total of r red balls have been withdrawn (here r n).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balls are withdrawn. Hint: if k balls are withdrawn in total, what color is the last ball?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many of each color are there in the other k 1 withdrawn balls?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