If the lifetime for a type of lamps follows a distribution with density function f(x) =12x²(1-x),0≤x≤1 , where is the lifetime in years. A sample of 100 units is selected randomly what is probability that its life time less than 7.5 month.

Suppose that a positive continuous random variable X has probability density function given by f(x)=c(14-5x^2), where c is a constant. What is the variance of this random variable? TYPE YOUR ANSWER AS THE NEAREST NUMBER WITH TWO DECIMAL DIGITS.

The diameter of a particle of contamination (in micrometers) is modeled with the probability density function 2 f(x)= = x3 for x > 1 . What is the probability that the diameter of the particle of contamination is below 2 micrometers?

The adult body lengths of deer mice are known to vary approximately Normally, with mean µ = 86 mm and standard deviation o = 8 mm. A researcher uses a random sample of 1 deer mice. What is the probability that the mice's body length is less than 80 mm? (i) Does the random variable represent Хor x? (ii) Draw a Normal density curve. (Label the horizontal axis and shade the area that the question is asking.) (iii) Draw a standard Normal density curve. (Label the horizontal axis and shade the area that the question is asking.)