For the bivariate random variable (X,Y), let the joint probability density function be \( f(x, y) = \frac{9(1 + x + y)}{2(1 + x)^4(1 + y)^5} \) for \( 0 < x < \infty, 0 < y < \infty \). The marginal pdf of X is:
\(\frac{3(2x+3)}{4(1+x)^4}\)
To find the marginal pdf of X, we integrate the joint pdf over all possible values of Y:
\[ f_X(x) = \int_{0}^{\infty} f(x,y) \, dy = \int_{0}^{\infty} \frac{9(1 + x + y)}{2(1 + x)^4(1 + y)^4} \, dy \]
Factor out terms not involving y:
\[ f_X(x) = \frac{9}{2(1 + x)^4} \int_{0}^{\infty} \frac{1 + x + y}{(1 + y)^4} \, dy \]
Split the integral:
\[ = \frac{9}{2(1 + x)^4} \left[ (1 + x) \int_{0}^{\infty} \frac{1}{(1 + y)^4} dy + \int_{0}^{\infty} \frac{y}{(1 + y)^4} dy \right] \]
Compute each integral separately:
1. \(\int \frac{1}{(1 + y)^4} dy = -\frac{1}{3(1 + y)^3}\)
Evaluated from 0 to ∞: \(0 - (-\frac{1}{3}) = \frac{1}{3}\)
2. \(\int \frac{y}{(1 + y)^4} dy\) (use substitution u = 1 + y):
\(= \int \frac{u-1}{u^4} du = \int (u^{-3} - u^{-4}) du = -\frac{1}{2u^2} + \frac{1}{3u^3}\)
Evaluated from 1 to ∞: \((0 + 0) - (-\frac{1}{2} + \frac{1}{3}) = \frac{1}{6}\)
Combine results:
\[ f_X(x) = \frac{9}{2(1 + x)^4} \left[ (1 + x)\cdot\frac{1}{3} + \frac{1}{6} \right] \]
\[ = \frac{9}{2(1 + x)^4} \left[ \frac{1 + x}{3} + \frac{1}{6} \right] \]
\[ = \frac{9}{2(1 + x)^4} \cdot \frac{2 + 2x + 1}{6} \]
\[ = \frac{9}{2(1 + x)^4} \cdot \frac{3 + 2x}{6} \]
\[ = \frac{9(3 + 2x)}{12(1 + x)^4} \]
\[ = \frac{3(3 + 2x)}{4(1 + x)^4} \]
\[ = \frac{3(2x + 3)}{4(1 + x)^4} \]
The correct answer is option 4: \(\frac{3(2x + 3)}{4(1 + x)^4}\)
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Which of the following are the properties of Binomial distribution?
A. May be symmetrical or skewed
B. Uni-modal, bell-shaped and symmetrical
C. Asymptotic to the x-axis
D. n and p are the two parameters
E. μ and σ are the two parameters
Choose thecorrectanswer from the options given below:
The following two statements relate to probability distributions. Choose the correct code for the statements being correct or incorrect.
Statement I: When ‘p' and 'q' are equal in the binomial distribution, the shape of the distribution is perfectly symmetrical irrespective of the size of 'n'.
Statement II: The mean and the variance of Poisson distribution are not equal.
Which of the following are the properties of normal distribution?
(A) May be symmetrical or skewed
(B) Uni-modal, bell-shaped and symmetrical
(C) Asymptotic to the x-axis
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(E) μ and σ are the two parameters
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A discrete random variable X has the following probability distribution.
X 1 2 3 4 5 6 7 P(X) K 2K 2K 3K K 2 2K 2 7K 2 + K What is the value of K?
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