Burrow Labs has taken on other startups in the home safety and security space and has recently started marketing a new smoke detector. Burrow's smoke detectors rely on C02 sensors that eventually fail, and that failure time dictates the average product lifetime of the smoke detector. Burrow manufactures three quarters of its smoke detectors in central Idaho, and the rest are manufactured in suburban Maine. Any single smoke detector's product lifetime can be modeled as a Exponential random variable. Each of the two locations sources its C02 sensors from different suppliers, so the smoke detectors manufactured in Maine have an average product lifetime of 7 years and the smoke detectors manufactured in Idaho have an average product lifetime of 6 years. All smoke detectors are sold online, so aside from the fact that a smoke detector is three times more likely to ship from the Idaho facility, you can't tell by looking at a single smoke detector where it was manufactured. Let T model the amount of time that passes until the C02 sensor (and therefore the smoke detector) fails, and let M be a discrete random variable that takes on the value of 1 for a smoke detector manufactured in Maine, and 0 otherwise.

Present the cumulative distribution and probability density functions for the random variable T. Both your CDF and your PDF should be analytic functions on t.