Binomial (m, p) B3. (a) Consider two independent random variables X₁ and X₂, with X₁ and X₂ Binomial (n, q). Find the distribution of X₁ + X₂ using the Moment Generating Functions (MGFs). Hint: Z~ Binomial(n, p) has MGF Mz(t) = (pe² + 1 - p)" (b) If X and Y are independent random variables ~ Exponential(2), use moment generating functions to obtain the distribution of U=X+Y. Hint: Z Exponential (A) has MGF Mz(t) = for t < A, = (A)") (c) For a random variable X we know that the MGF is: H~ Gamma(A, a) has MGF Mu(t)= = 3 Mx (t) = 3 ³ 1 Find the distribution of X. (d) Find the MGF for the continuous random variable X~ Uniform(a, b). Show your work. (e) Find the MGF for the discrete random variable X with probability mass function: Px (k) 1 H for t < 3. 2 k = 1 k=0 k = -1
Binomial (m, p) B3. (a) Consider two independent random variables X₁ and X₂, with X₁ and X₂ Binomial (n, q). Find the distribution of X₁ + X₂ using the Moment Generating Functions (MGFs). Hint: Z~ Binomial(n, p) has MGF Mz(t) = (pe² + 1 - p)" (b) If X and Y are independent random variables ~ Exponential(2), use moment generating functions to obtain the distribution of U=X+Y. Hint: Z Exponential (A) has MGF Mz(t) = for t < A, = (A)") (c) For a random variable X we know that the MGF is: H~ Gamma(A, a) has MGF Mu(t)= = 3 Mx (t) = 3 ³ 1 Find the distribution of X. (d) Find the MGF for the continuous random variable X~ Uniform(a, b). Show your work. (e) Find the MGF for the discrete random variable X with probability mass function: Px (k) 1 H for t < 3. 2 k = 1 k=0 k = -1
A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
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Transcribed Image Text:B3. (a) Consider two independent random variables X₁ and X₂, with X₁ ~ Binomial (m, p)
and X₂
Binomial(n, q). Find the distribution of X₁ + X2 using the Moment
Generating Functions (MGFs).
~
Hint: Z~ Binomial(n, p) has MGF Mz(t) = (pet +1-
- p)")
(b) If X and Y are independent random variables ~ Exponential(2), use moment
generating functions to obtain the distribution of U=X+Y.
Hint: Z~ Exponential (A) has MGF Mz(t) =
for t < >,
H ~
Gamma(X, a) has MGF MH(t) = (+)ª)
(c) For a random variable X we know that the MGF is:
3
3-t
Mx (t)
=
Find the distribution of X.
(d) Find the MGF for the continuous random variable X~ Uniform(a, b). Show your
work.
(e) Find the MGF for the discrete random variable X with probability mass function:
{
=
Px (k):
for t < 3.
1
k = 1
k = 0
k = -1
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Step 1: Write the given information.
VIEWStep 2: Determine the distribution of X_1 and X_2 using the Moment Generating functions.
VIEWStep 3: Determine the distribution of U = X+Y.
VIEWStep 4: Determine the distribution of X.
VIEWStep 5: Determine the MGF for the continuous random variable X~ Uniform(a, b)
VIEWStep 6: Determine the MGF for the discrete random variable X with probability mass function.
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