Let the joint probability density function of \( (X, Y) \) be \[f(x, y) = \begin{cases} 6xy^2 & \text{if } 0 < x < 1, 0 < y < 1 \\ 0, & \text{otherwise} \end{cases}\] Then \( P\left(\frac{1}{2} < X < \frac{3}{4}\right) \) is:
5/16
To calculate P(1/2 < X < 3/4), we integrate the joint density function over the desired range of X:
P(1/2 < X < 3/4) = ∫[1/2 to 3/4] ∫[0 to 1] 6xy² dy dx.
First, integrate with respect to y:
∫[0 to 1] 6xy² dy = 6x[y³ / 3] from 0 to 1 = 6x / 3 = 2x.
Now integrate with respect to x:
∫[1/2 to 3/4] 2x dx = [x²] from 1/2 to 3/4 = (3/4)² - (1/2)² = 9/16 - 1/4 = 5/16.
Thus, the answer is 5/16.
Similar Questions
Let X and Y have the joint p.m.f. f(x, y) = x + y / 21, where x = 1, 2, 3 and y = 1, 2. The marginal p.m.f. of X is:
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\[ f(x) = \frac{n!}{x!(n-x)!}k^{x}(1-k)^{n-x}, \quad x = 0,1,2,\ldots,n \]
Let \( X \) have pdf:
\[ f(x) = \begin{cases} \frac{3(2x - x^2)}{4} & \text{for } 0 \leq x \leq 2, \\ 0, & \text{otherwise} \end{cases} \]
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Important Questions from Probability Distribution
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?
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
(D) m and p are the two parameters
(E) μ and σ are the two parameters
Choose the correct answer from the options given below:
Let X and Y have the joint p.m.f. f(x, y) = x + y / 21, where x = 1, 2, 3 and y = 1, 2. The marginal p.m.f. of X is:
