A random variable X is distributed at random between the values 0 and 1 in such a way that the PDF of X is f(x) = x2(1 - x3), where k is a constant. The value of k is:
6
To ensure that f(x) is a valid probability density function (PDF), we integrate f(x) over the interval [0, 1] and set the integral equal to 1. Solving this gives the value of k as 6.
Similar Questions
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:
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:
For a random variable X following Poisson distribution with parameter 5, the variance of X is:
If \( n \) is a natural number, then, for what value of \( k \) is the following function a probability mass function?
\[ f(x) = \frac{n!}{x!(n-x)!}k^{x}(1-k)^{n-x}, \quad x = 0,1,2,\ldots,n \]
Let \( X \) have pdf:
\[ f(x) = \begin{cases} \frac{3(2x - x^2)}{4} & \text{for } 0 \leq x \leq 2, \\ 0, & \text{otherwise} \end{cases} \]
Then the mode is equal to:
If the first two raw moments of X are equal to zero, then P(X = 0) is equal to:
Let the probability density function of \( X \) be:
\[ f(x) = \begin{cases} 3(1-x)^2 & \text{for } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases} \]
Then \( P\left(\frac{1}{2} < X < \frac{3}{4}\right) \) is equal to:
Let the probability mass function of \( X \) be given by:
\[ p(x) = c \frac{4^x}{x!}, \quad x = 0,1,2,3, \ldots \]Then the constant \( c \) is equal to:
In an industry, the risk of suffering from occupational disease is 20%. The probability that out of 6 workers, four will suffer from the disease is:
If X has the probability density function:
\( f(x) = \begin{cases} \frac{1}{4}, & 1 \leq x \leq 5, \\ 0, & \text{otherwise} \end{cases} \)Then the median is:
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 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:
