Final answer:
The mean (μ) and standard deviation (σ) of X with the probability density function f(x) = λe2λ|x| for a given λ=0.25 are both 4. Probabilities regarding the number of literate people in a sample can be calculated using the exponential distribution's properties.
Step-by-step explanation:
To find the mean and variance of a random variable X with a given probability density function (pdf), we integrate the pdf to obtain the expected value (mean) and then integrate the squared deviation from the mean to obtain the variance. The pdf provided is f(x) = λe2λ|x|, which suggests that X follows an exponential distribution with the parameter λ. For an exponential distribution Exp(λ), the mean (μ) and variance (σ2) are 1/λ and 1/λ2 respectively.
The given distribution notation is X~ Exp(0.25) which implies that λ=0.25. Using the properties of an exponential distribution, the mean (μ) would be 1/λ = 4, and the variance (σ2) would also be 1/λ2 = 16. Since the standard deviation is the square root of the variance, the standard deviation (σ) would be √16 = 4, which is the same as the mean in this case.
The probability that more than five people in the sample are literate or the probability that three or four people are literate can be found by integrating the pdf over the corresponding intervals, using the cumulative distribution function