f(x₁, x₂) = {1,- ([1-a(1-2e-x₁)(1-2e-x₂)]ex₁-x2 otherwise ,0 ≤ x₁,0 ≤ x₂. 0, i) Find the marginal distribution of X₁. ii) Find E(X₁X₂). b) Consider a random variable K with parameter p, whose probability mass function (PMF) is given by f(k) = {* elsewhere k=1,2,. lo, i) Derive the moment generating function of K. ii) Use the result obtained in i), to find the expected value of K. iii) Use the result obtained in i), to find the variance of value of K. c) Mr Ledwaba from Department of Health assumed that a random variable X is normal with mean 2 and standard deviation 3 and that random variable Y is normal with mean 0 and standard deviation 4. Suppose that X and Y are independent, use an appropriate method to determine his probability distribution of the random variable X + Y.

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a) Suppose that, for −1≤ a ≤ 1, the probability density function of (X₁, X₂) is given by
f(x₁, x₂) = {11-
([1-α(1 - 2e-x₁)(1-2e-x²)]ex1-x2
otherwise
,0 ≤ x₁,0 ≤ x₂.
i) Find the marginal distribution of X₁.
ii) Find E(X₁X₂).
b) Consider a random variable K with parameter p, whose probability mass function (PMF) is
given by
f(x) = { pak k = 1.,2., .......
elsewhere
i) Derive the moment generating function of K.
ii) Use the result obtained in i), to find the expected value of K.
iii) Use the result obtained in i), to find the variance of value of K.
c) Mr Ledwaba from Department of Health assumed that a random variable X is normal with
mean 2 and standard deviation 3 and that random variable Y is normal with mean 0 and
standard deviation 4. Suppose that X and Y are independent, use an appropriate method
to determine his probability distribution of the random variable X + Y.
Transcribed Image Text:a) Suppose that, for −1≤ a ≤ 1, the probability density function of (X₁, X₂) is given by f(x₁, x₂) = {11- ([1-α(1 - 2e-x₁)(1-2e-x²)]ex1-x2 otherwise ,0 ≤ x₁,0 ≤ x₂. i) Find the marginal distribution of X₁. ii) Find E(X₁X₂). b) Consider a random variable K with parameter p, whose probability mass function (PMF) is given by f(x) = { pak k = 1.,2., ....... elsewhere i) Derive the moment generating function of K. ii) Use the result obtained in i), to find the expected value of K. iii) Use the result obtained in i), to find the variance of value of K. c) Mr Ledwaba from Department of Health assumed that a random variable X is normal with mean 2 and standard deviation 3 and that random variable Y is normal with mean 0 and standard deviation 4. Suppose that X and Y are independent, use an appropriate method to determine his probability distribution of the random variable X + Y.
a) Suppose that, for −1≤ a ≤ 1, the probability density function of (X₁, X₂) is given by
f(x₁, x₂) = {11-
([1-α(1 - 2e-x₁)(1-2e-x²)]ex1-x2
otherwise
,0 ≤ x₁,0 ≤ x₂.
i) Find the marginal distribution of X₁.
ii) Find E(X₁X₂).
b) Consider a random variable K with parameter p, whose probability mass function (PMF) is
given by
f(x) = { pak k = 1.,2., .......
elsewhere
i) Derive the moment generating function of K.
ii) Use the result obtained in i), to find the expected value of K.
iii) Use the result obtained in i), to find the variance of value of K.
c) Mr Ledwaba from Department of Health assumed that a random variable X is normal with
mean 2 and standard deviation 3 and that random variable Y is normal with mean 0 and
standard deviation 4. Suppose that X and Y are independent, use an appropriate method
to determine his probability distribution of the random variable X + Y.
Transcribed Image Text:a) Suppose that, for −1≤ a ≤ 1, the probability density function of (X₁, X₂) is given by f(x₁, x₂) = {11- ([1-α(1 - 2e-x₁)(1-2e-x²)]ex1-x2 otherwise ,0 ≤ x₁,0 ≤ x₂. i) Find the marginal distribution of X₁. ii) Find E(X₁X₂). b) Consider a random variable K with parameter p, whose probability mass function (PMF) is given by f(x) = { pak k = 1.,2., ....... elsewhere i) Derive the moment generating function of K. ii) Use the result obtained in i), to find the expected value of K. iii) Use the result obtained in i), to find the variance of value of K. c) Mr Ledwaba from Department of Health assumed that a random variable X is normal with mean 2 and standard deviation 3 and that random variable Y is normal with mean 0 and standard deviation 4. Suppose that X and Y are independent, use an appropriate method to determine his probability distribution of the random variable X + Y.
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