MATH2560H Challenge Problem 01 Probability Distributions
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Challenge Problem 01
Probability Distributions
1 Probability Density Functions
The length of time to failure (in hundreds of hours) for a transistor is a random variable Y with distribution function given by:
(a) Show that F(y) has the properties of a distribution function.
(b) Find the 0.30-quantile (or 30%), ϕ0 .30 , of Y.
(c) Find f(y).
(d) Find the probability that the transistor operates for at least 200 hours. (e) Find P‘Y > 100 'Y ≤ 200) .
2 Cumulative Distribution Functions
Suppose that the continuous random variable X has the distribution fX (x), −∞ < x < ∞, which is symmetric about the value x = 0. Evaluate the integral:
where FX (t) is the CDF for X, and k is a non-negative real number.
Hint: Use integration by parts, and recall the relationship between fX (x) and FX (t).
Specifications for Success
To obtain satisfactory on this assignment, all of the criteria below must be met
• All problems are attempted.
• Concluding statements are present where appropriate.
• Correct methodology described in Q1.
• Mostly correct application of methodology to (a) through (c) in Q1.
• Derivation of numerical conditional probability in Q1 (a) correct.
• Explicit expression derived in Q2 (b) correct.
2023-02-22