Probability mass function and joint density

May 2008
12
0
1.A receiving depot receives a shipment of 100 generators, 5 of which are defective.
Four are selected at random without replacement for inspection. Let​
Y

be the number of defectives.
(a) What is the probability mass function of​
Y ?

(b) Calculate
P(1 Y 3).
2. Suppose the density of X is given by
\(\displaystyle
f(x)=\frac{1}{4} xe^{-\frac{x}{2}} \quad x>0
\)
0 otherwise
Use the moment generating function to calculate Var (X).

3.
Let​
X and Y be independent normal random variables, each having parameters μ and σ2.
(a) Compute the joint density of
A = X + Y and B = X Y .

(b) Are
A and B independent?
 
Last edited:

mr fantastic

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1.A receiving depot receives a shipment of 100 generators, 5 of which are defective.

Four are selected at random without replacement for inspection. Let
Y

be the number of defectives.
(a) What is the probability mass function of Y ?


(b) Calculate P(1 Y 3).


[snip]
(a) Review the hypergeometric distribution.

(b) Pr(Y ≤ 3) - Pr(Y = 0).

[snip]
2. Suppose the density of X is given by
\(\displaystyle
f(x)=\frac{1}{4} xe^{-\frac{x}{2}} \quad x>0
\)
0 otherwise
Use the moment generating function to calculate Var (X).


[snip]
Apply the definition of the mgf. Use integration by parts to do the required integration. If you need more help, please show all yuor working and say where you are stuck.

[snip]
3.

Let


X and Y be independent normal random variables, each having parameters μ and σ2.

(a) Compute the joint density of A = X + Y and B = X Y .

(b) Are A and B independent?
(a) Use the Change of Variable Theorem. I assume you have been taught it. Where are you stuck?
(b) Can f(a, b) be written in the form g(a) h(b)?