STM2EPS - Electrical Engineering Assignment Help

Internal Code: 1AJJFD

STM2EPS - Electrical Engineering Assignment

1. Suppose that a satellite transmits (sends) a signal that consists of either a 0 or 1. Let T0 denote the event that the satellite transmits a 0, and T1 denote the event that the satellite transmits a 1. The signal is received on Earth by a ‘receiving dish’. Let R0 and R1 denote the events that a 0 and a 1 are received respectively. Suppose that the probability that a 1 is transmitted is 0.95 and that there is a 5% change that a transmitted 1 will be incorrectly received as a 0. The probability that a transmitted 0 is received as a 0 is 0.99. Note that if a 1 is not transmitted then a 0 is transmitted, and vice versa. Similarly, if a 1 is not received then a 0 is received, and vice versa. In parts (1a) to (1d) of this question, you are being asked for a numerical answer and not a description of what the respective probabilities represent. (a) What is P(T0) and P(T1)? (b) What is P(R0|T0) and P(R0|T1)? (c) Using the Law of Total Probability, calculate P(R0)? (d) Now, if a 0 is received, what is the probability that a 0 was sent? (e) Suppose that a 0 is received, are you confident that a 0 was transmitted? Justify your answer with one of your answers from above. 2. Let T denote a random variable for the lifetime, measured in months, of an electrical component. The probability density function for T is f(t) = ( ce?t/2, t ? 0 0, otherwise (1) for some value of c > 0. If you are instructed to use R, then you must include your R commands in your submission. (a) For f to be a valid probability density function, show clearly that c must be equal to 1/2. Use c = 1/2 for the remainder of the question. (b) Create a plot of the probability density function using R. The plot should have adequate labels and the scale for the horizontal axis should be chosen appropriately. (c) Using integration by parts, show clearly that Z te?t/2 dt = ?2(2 + t)e ?t/2 . (2)You may find the following compact definition useful: Ru dv = uv ? Rv du. (d) Using equation (2) above, calculate E(T). (e) Now use the R function integrate to check your answer to 2d. (f) Using the fact that Z ?0 1 2 t 2exp(?t/2)dt = 8, calculate Var(T). What units is it measured in? (g) For t ? [0, ?], show that the probability distribution function of T (i.e. F(t) = P(T ? t)) is F(t) = 1 ? e ?t/2 (h) Calculate the median lifetime of the electrical component. (i) Recall that R(t) = P(T > t) denotes the reliability of the component at time t. Calculate R(6). (j) Suppose that a system will only function if this component operates. It is decided a number of these components will be used in parallel to increase the probability that the system does not fail prior to the end of its warranty period of 6 months. How many components need to be arranged in parallel for the probability of the system to still be functioning at 6 months to be at least 0.95? 3. Let X be the number of cars detected by a pressure plate starting to cross a small bridge in a 10 second interval. Let Y denote the actual number of cars starting to cross the bridge in the 10 second interval. The joint probability mass function for (X, Y ) is below. p(x, y) x = 0 x = 1 x = 2 y = 0 0.5 0.0 0.0 y = 1 0.05 0.3 0.0 y = 2 0.03 0.02 0.1 (a) What is P(X = 1 and Y = 2) and what does this probability represent? (b) Derive the marginal probability mass function for the random variable X. (c) Similarly, derive the marginal probability mass function for the random variable Y . (d) Given that exactly one car starts crossing bridge, what is the probability that the pressure plate incorrectly detects no cars? (e) Given that exactly one car starts crossing bridge, what is the probability that the pressure plate incorrectly detects two cars? (f) If exactly one car starts crossing the bridge, is pressure plate more likely to under-detect (i.e. detect fewer) or over-detect the number of cars? Explain.