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$ \checkmark $ Question Three (a) Given a Poisson distribution such that $ P(X=2)=P(X=3) $. Find $ E(x) $ (b) The mean and variance of a binomial distribution are 4 and $ \frac{4}{3} $ respectively. Find $ n $ and $ p $ (c) Fourteen applicants apply for the post of a marketer in a company. 8 of them have university degrees. If 5 of these applicants are randomly selected for an interview, what is the probability that 3 of them have university degrees? (d) The hardness of a certain alloy (measured on Rockwell scale) is a random variable $ X $. Given that $ X \sim $ Uniform $ [50,80] $. Find i. $ \mathrm{E}(\mathrm{X}) $ ii. $ \quad \operatorname{Var}(X) $ (16Marks) Question Four (a) Highlight any three features of a Binomial distribution (b) Derive the mean and variance of a Binomial distribution (c) If $ 40 \% $ of cocoa seeds bought by a produce buyer are defective, find the probability that out of 5 cocoa seeds selected at random that at most 2 are defective. (d) If the probability that any marriage contracted in a local government collapses is 0.001 , determine the probability that out of 4000 such marriages exactly 3 will collapse. Question Five - (a) Given that $ E(Y+4)=10 $ and $ E\left[(Y+4)^{2}\right]=116 $. Find: i. $ \operatorname{Var}(Y+4) $ ii. $ E(Y) $ (b) Given that $ \Gamma(c:)=\int_{0}^{\infty} x^{\alpha-1} e^{-x} d x $. Evaluate the following: (16Marks) 4000 r. $ 301=4 $ ii. $ \int_{0}^{\infty} x^{4} e^{-x} d x $ (c) Given that $ X \sim $ Exponential $ (\theta) $, using information in $ 5(\mathrm{~b}) $ or otherwise, Find the: i. ii. $ \operatorname{Var}(X) $ (16Marks) Question Six (a) Let $ X \sim $ Poisson ( $ \lambda $ ). i. Find the Moment Generating Function (MFG) of $ X $ ii. Use your result in a(i) above to obtain the mean and variance of $ X $ (b) Consider a sequence of independent rolls of a die. Find the probability that the first 5 is observed on the 6th trial (c) Let $ X $ and $ Y $ be two jointly continuous random variable with the joint PDF 16 \[ f(x)=\left\{\begin{array}{rc} x+c y^{2}, & 1 \leq x \leq 1,0 \leq y \leq 1 \\ 0, & \text { otherwise } \end{array}\right. \] Find the constant $ c $ (16Marks)

          \( \checkmark \) Question Three
(a) Given a Poisson distribution such that \( P(X=2)=P(X=3) \). Find \( E(x) \)
(b) The mean and variance of a binomial distribution are 4 and \( \frac{4}{3} \) respectively. Find \( n \) and \( p \)
(c) Fourteen applicants apply for the post of a marketer in a company. 8 of them have university degrees. If 5 of these applicants are randomly selected for an interview, what is the probability that 3 of them have university degrees?
(d) The hardness of a certain alloy (measured on Rockwell scale) is a random variable \( X \). Given that \( X \sim \) Uniform \( [50,80] \). Find
i. \( \mathrm{E}(\mathrm{X}) \)
ii. \( \quad \operatorname{Var}(X) \)
(16Marks)

Question Four
(a) Highlight any three features of a Binomial distribution
(b) Derive the mean and variance of a Binomial distribution
(c) If \( 40 \% \) of cocoa seeds bought by a produce buyer are defective, find the probability that out of 5 cocoa seeds selected at random that at most 2 are defective.
(d) If the probability that any marriage contracted in a local government collapses is 0.001 , determine the probability that out of 4000 such marriages exactly 3 will collapse.
Question Five
- (a) Given that \( E(Y+4)=10 \) and \( E\left[(Y+4)^{2}\right]=116 \). Find:
i. \( \operatorname{Var}(Y+4) \)
ii. \( E(Y) \)
(b) Given that \( \Gamma(c:)=\int_{0}^{\infty} x^{\alpha-1} e^{-x} d x \). Evaluate the following:
(16Marks) 4000 r. \( 301=4 \)
ii. \( \int_{0}^{\infty} x^{4} e^{-x} d x \)
(c) Given that \( X \sim \) Exponential \( (\theta) \), using information in \( 5(\mathrm{~b}) \) or otherwise, Find the:
i.
ii.
\( \operatorname{Var}(X) \)
(16Marks)
Question Six
(a) Let \( X \sim \) Poisson ( \( \lambda \) ).
i. Find the Moment Generating Function (MFG) of \( X \)
ii. Use your result in a(i) above to obtain the mean and variance of \( X \)
(b) Consider a sequence of independent rolls of a die. Find the probability that the first 5 is observed on the 6th trial
(c) Let \( X \) and \( Y \) be two jointly continuous random variable with the joint PDF
16
\[
f(x)=\left\{\begin{array}{rc}
x+c y^{2}, & 1 \leq x \leq 1,0 \leq y \leq 1 \\
0, & \text { otherwise }
\end{array}\right.
\]

Find the constant \( c \)
(16Marks)

$Question Three (a) Given a Poisson distribution such that P(X=2)=P(X=3). Find E(x) (b) The mean and variance of a binomial distribution are 4 and (4)/(3) respectively. Find n and p (c) Fourteen applicants apply for the post of a marketer in a company. 8 of them have university degrees. If 5 of these applicants are randomly selected for an interview, what is the probability that 3 of them have university degrees? (d) The hardness of a certain alloy (measured on Rockwell scale) is a random variable X. Given that X ∼ Uniform [50,80]. Find i. E(X) ii. Var(X) (16Marks) Question Four (a) Highlight any three features of a Binomial distribution (b) Derive the mean and variance of a Binomial distribution (c) If 40 % of cocoa seeds bought by a produce buyer are defective, find the probability that out of 5 cocoa seeds selected at random that at most 2 are defective. (d) If the probability that any marriage contracted in a local government collapses is 0.001 , determine the probability that out of 4000 such marriages exactly 3 will collapse. Question Five - (a) Given that E(Y+4)=10 and E[(Y+4)^2]=116. Find: i. Var(Y+4) ii. E(Y) (b) Given that Γ(c:)=∫0^∞ x^α-1 e^-x d x. Evaluate the following: (16Marks) 4000 r. 301=4 ii. ∫0^∞ x^4 e^-x d x (c) Given that X ∼ Exponential (θ), using information in 5( b) or otherwise, Find the: i. ii. Var(X) (16Marks) Question Six (a) Let X ∼ Poisson ( λ ). i. Find the Moment Generating Function (MFG) of X ii. Use your result in a(i) above to obtain the mean and variance of X (b) Consider a sequence of independent rolls of a die. Find the probability that the first 5 is observed on the 6th trial (c) Let X and Y be two jointly continuous random variable with the joint PDF 16 f(x)={ x+c y^2, 1 ≤ x ≤ 1,0 ≤ y ≤ 1 0, otherwise . Find the constant c (16Marks)$

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