Suppose $X$ and $Y$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability that $X \ge Y$ is _________
Probability P(X ≥ Y) Calculation
We are given two random variables, X and Y, which are independent and identically distributed (i.i.d.). Both follow a uniform distribution on the interval $[0, 1]$, denoted as X, Y ~ U[0, 1]. We need to find the probability P(X ≥ Y).
Geometric Interpretation
The joint distribution of X and Y is uniform over the unit square defined by $0 \le x \le 1$ and $0 \le y \le 1$. The total area of this sample space is $1 \times 1 = 1$.
The condition X ≥ Y corresponds to the region within the unit square where the x-coordinate is greater than or equal to the y-coordinate. This region is a triangle with vertices at (0, 0), (1, 0), and (1, 1).
Calculating the Probability
The area of this favorable region (the triangle) is calculated as:
Area = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}$.
Since the joint distribution is uniform, the probability of the event X ≥ Y is the ratio of the favorable area to the total area of the sample space:
P(X ≥ Y) = $\frac{\text{Area where } X \ge Y}{\text{Total Area}} = \frac{1/2}{1} = \frac{1}{2}$.
Alternatively, due to symmetry (since X and Y are i.i.d. from the same distribution), the probability P(X > Y) must be equal to P(Y > X). Since P(X=Y) = 0 for continuous distributions, we have P(X ≥ Y) = P(X > Y). Also, P(X > Y) + P(Y > X) + P(X=Y) = 1. This simplifies to 2 * P(X > Y) = 1, so P(X > Y) = 1/2. Therefore, P(X ≥ Y) = 1/2.
Important Questions from Basics of Probability
Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is ______ . (rounded off to three decimal places)
- A quadratic polynomial $(x - \alpha)(x - \beta)$ over complex numbers is said to be square invariant if $(x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2)$. Suppose from the set of all square invariant quadratic polynomials we choose one at random.
The probability that the roots of the chosen polynomial are equal is ____________ (rounded off to one decimal place) - If, out of 50 fish of a pond, 12 had no ectoparasite and remaining had varying numbers of parasites on them, then how many of fish had at least one parasite ?
For a distribution with mean, median, mode and standard deviation 25, 24, 26 and 5 respectively, Karl Pearson's coefficient of skewness equals to:
Suppose two distributions have the same mean, same standard deviation and are equally skewed, but the first distribution is more peaked than the
other. This feature of the frequency distributions in which they are different are called:
