Example Question - random variable
Here are examples of questions we've helped users solve.
Understanding the Binomial Distribution in Probability Theory
The image contains a question in French about probability. The question translates to: "We know from experience that a certain surgical operation has a 95% chance of success. We are about to perform this operation on 6 patients. Let X be the random variable equal to the number of successful operations out of the 6 attempts. 1) What is the law followed by X?" The law followed by X is the binomial distribution. This is because the binomial distribution is applicable for a fixed number of independent trials (in this case, 6), where each trial has only two possible outcomes (success or failure), and the probability of success is the same for each trial. In mathematical terms, if \( p \) is the probability of success for each trial, \( n \) is the number of trials, and \( k \) is the number of successes, the probability \( P(X = k) \) is given by: \[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \] For this particular case: - \( n = 6 \) (the number of patients), - \( p = 0.95 \) (the probability of success for the surgical operation), - \( k \) can be any integer from 0 to 6 (the number of successful operations). So the random variable \( X \) representing the number of successful operations follows a binomial distribution with parameters \( n = 6 \) and \( p = 0.95 \).
Probability Calculation Using Probability Density Function
The image shows a graph with a density function \( f(x) \) for a random variable X and asks to find the probability that X is between 1 and 4, \( P(1 < X < 4) \). The graph depicts a triangle that represents the probability density function, with the base of the triangle spanning from x = 0 to x = 4 and the height at f(x) = 1/2 at x = 0. The area under the curve of a probability density function \( f(x) \) between two points a and b gives the probability \( P(a < X < b) \). To find \( P(1 < X < 4) \), we will calculate the area of the triangle between x = 1 and x = 4. The original triangle's base is 4 units long (from x = 0 to x = 4), and the height is \( f(x) = 1/2 \). The area of the full triangle is: \( \text{Area} = \frac{1}{2} (\text{base}) (\text{height}) = \frac{1}{2} (4) \left(\frac{1}{2}\right) = 1 \) Now, we need only the area from x = 1 to x = 4. To find this portion, we subtract the area of the smaller triangle from x = 0 to x = 1 from the total area. The smaller triangle has a base of 1 unit and the same height of \( f(x) = 1/2 \). The area of this smaller triangle is: \( \text{Area}_{\text{small}} = \frac{1}{2} (\text{base of small triangle}) (\text{height}) = \frac{1}{2} (1) \left(\frac{1}{2}\right) = \frac{1}{4} \) Now subtract the area of the small triangle from the total area to find the desired probability: \( P(1 < X < 4) = \text{Area}_{\text{total}} - \text{Area}_{\text{small}} = 1 - \frac{1}{4} = \frac{3}{4} \) So, the probability that X is between 1 and 4 is \( \frac{3}{4} \) or 0.75.
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