STA219: Probability and Statistics for Engineering Assignment 5 with Solution

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1. (10 points) 𝑋 and 𝑌 are independent random variables, and 𝑋~𝐸𝑥𝑝(𝜆), 𝑌~𝐸𝑥𝑝(𝜇). Define
random variable 𝑍 as follows:
𝑍 = .
1, 𝑖𝑓 𝑋 ≤ 𝑌,
0, 𝑖𝑓 𝑋 > 𝑌.
Derive the PMF of Z.
2. (10 points) Suppose 𝑋 and 𝑌 are independent, identically distributed random variables, and
follow geometric distribution, which is 𝑃(𝑋 = 𝑘) = (1 − 𝑝)!”#𝑝, 𝑘 = 1,2, …. Derive the PMF
of 𝑍 = max (𝑋, 𝑌).
3. (10 points) Consider a device equipped with 3 identical electrical components that operate
independently. The operating times of these components follow an exponential distribution
with parameter 𝜆. The device functions normally only when all 3 components are working
properly. Please find the PDF of the operating time 𝑇 of the device.
4. (20 points) Suppose 𝑋 and 𝑌 are bivariate normally distributed with means 𝜇$ = 𝜇% = 0,
variances 𝜎$
& = 𝜎%
& = 1, and correlation coefficient 𝜌.
(1) Please derive the distribution and PDF of 𝑋 − 𝑌. (10 points)
(2) Please calculate the covariance and correlation coefficient of 𝑋 − 𝑌 and 𝑋𝑌. (10 points)
5. (15 points) The PDF of zero mean bivariate normal random vector (𝑋, 𝑌)’ is of the form
𝑓(𝑥, 𝑦) = 1
2𝜋𝜎$𝜎%D1 − 𝜌& 𝑒𝑥𝑝 F− 1
2(1 − 𝜌&)
G
𝑥&
𝜎$
& − 2𝜌
𝑥𝑦
𝜎$𝜎%
+ 𝑦&
𝜎%
&IJ.
(1) Please derive the marginal density function of Y. (5 points)
(2) Prove that the conditional PDF of 𝑋 given 𝑌 = 𝑦 is normal, and identify its conditional
mean and variance. (10 points)
(For both questions, please provide a detailed proof. Do not directly use the conclusion from
the slide.)
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6. (20 points) The annual revenues of Company X and Company Y are positively correlated since
the correlation coefficient between the two revenues is 0.65. The annual revenue of Company
X is, on average, 4500 with standard deviation 1500. The annual revenue of Company Y is, on
average, 5500 with standard deviation 2000. Assume that X and Y are bivariate normally
distributed.
(1) Calculate the probability that annual revenue of Company X is less than 6800 given that
the annual revenue of Company Y is 6800. (10 points)
(2) Calculate the probability that the annual revenue of Company X is greater than that of
Company Y given that their total revenue is 12000. (10 points)
7. (15 points) Since both the normal distribution and the Poisson distribution can be used to
approximate binomial distribution, can we use normal distribution to approximate the Poisson
distribution?
(1) First derive the normal approximation to the Poisson distribution 𝑃𝑜𝑖𝑠𝑠𝑜𝑛(𝜆). (5 points)
(2) Use Python to plot the Poisson distribution and its corresponding normal approximation
under different values of 𝜆 (similar to the plots on page 51 of the slide). Explain in which
cases the normal approximation works well and in which cases it does not work well. (10
points)
(Please provide both the plots and the code.)
(既然正态分布和泊松分布都能近似⼆项分布，那正态分布是不是也可以近似泊松分布？
请给出泊松分布𝑃𝑜𝑖𝑠𝑠𝑜𝑛(𝜆)的正态近似，并用 Python 代码画出 𝜆 取不同的值时与其近似
正态分布的图形(与 PPT 上类似的图)，说明什么情况下正态近似比较好，什么情况下不太
好。)