In certain experiments, the error made in determining the density of a substance is a random variable X, which is equally likely to take any value between two stated bounds -0.025 and 0.025.

(i) What distribution may be used to model X? Write down the probability density function of X. What is the probability that such an error will be between 0.010 and 0.015?

(ii) What is the probability that such an error will be between -0.012 and 0.012?

(iii) Determine the mean and variance of X.

Update:

I think the first one is geometric distribution, but i don't know how to calculate pdf of x

Relevance

The parameters you give for the probability density function of X do not tell us much. All we really have is the minimum and maximum. If X is equally likely to take any value on that interval, while the distribution is continuous, then that probability is zero.

One thing for certain, you cannot be looking for a geometric distribution. That distribution would be discrete, not continuous.

Below are two distinct probability mass functions satisfying the stated condition.

Here is a pdf for a uniform distribution:

f(x) = 20, -0.025 ≤ x ≤ 0.025

∫[-0.025,0.025] f(x) dx = 1

For any t on [-0.025, 0.025], P(X = t) = ∫[t,t] f(x) dx = 0.

P(0.010 ≤ X ≤ 0.025) = ∫[0.010,0.025] f(x) dx = 0.3

Here is a pdf for a non-uniform linear distribution:

g(x) = 20 - 800x, -0.025 ≤ x ≤ 0.025

∫[-0.025,0.025] g(x) dx = 1

For any t on [-0.025, 0.025], P(X = t) = ∫[t,t] g(x) dx = 0.

P(0.010 ≤ X ≤ 0.025) = ∫[0.010,0.025] g(x) dx = 0.09

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