Discrete Probability Distribution Explained
A discrete probability distribution is a fundamental concept in statistics used to describe the likelihood of different outcomes for a discrete random variable. A discrete random variable can only take on a finite number of distinct values or a countably infinite number of distinct values (e.g., number of heads in 3 coin flips: 0, 1, 2, 3).
Defining the Distribution
- Key Characteristic: The defining feature of a discrete probability distribution is that it provides a specific probability for each distinct value that the random variable can assume.
- Option 2 Correctness: Option 2, "assigns a probability to each possible value of the random variable," perfectly captures this definition. For every possible outcome (value), there is a corresponding probability value.
Why Other Options Are Incorrect
- Option 1: While a distribution lists possible values, simply listing them isn't enough. It must also include the associated probabilities.
- Option 3: Probabilities themselves must be between 0 and 1, inclusive. This option incorrectly suggests the *variable* assumes values between -1 and +1 and is poorly phrased.
- Option 4: Probability distributions are typically defined by, and therefore highly dependent on, their parameters (e.g., the parameter '$p$' in a binomial distribution).
In summary, a discrete probability distribution connects each possible distinct outcome of a random variable with its specific probability of occurrence.
Important Questions from Probability Distribution
If the mean and variance of a binomial distribution are 5 and 4, respectively, then the value of n is:
For the distribution with unknown θ
\(f(x,\theta ) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{\theta };0 \le x \le \theta }\\ {0;elsewhere} \end{array}} \right.\)
We set the testing of hypothesis H 0 ∶ θ = 1 vs H 1 ∶ θ = 2. When the critical region X ≥ 0.4, the value of probability of type-II error is:
For the cumulative distribution function \(F(x) = \left\{ {\begin{array}{*{20}{c}} {0;x < - 1}\\ {\frac{1}{2}{{(x + 1)}^2}; - 1 \le x < 0}\\ {1 - \frac{{{{(1 - x)}^2}}}{2};0 \le x < 1}\\ {1.1 < x < \infty } \end{array}} \right.\)
the upper quartile point is
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:
