experience my assignments @ MIT

experience my assignments @ MIT 3.1 Expanding, we have E[XTAX] = ... 3.2 We have Ln(μ) = ... 3.3 We are trying to minimize ... Statistics 3.4 The distribution of μ̂MLE ... 3.5 We have B̂MLE ... Notice that this is equivalent to V(B̂MLE) = ... 3.6 Since μ̂MLE = Xn, we will use the Multivariate Delta Method. We get ∇g(X) = ... 4.1 We have P(a) = ... 4.2 We have p(x1, x2) = ... 4.3 We have L1000(θ) = ... 5.1 If x < 0, we have fθ(x) = ... class VAE(nn.Module): def __init__(self): super().__init__() # FIX ME: Create two encoder layers self.encoder = nn.Sequential( nn.Linear(input_size, num_hidden), nn.ReLU(), nn.Linear(num_hidden, num_hidden), nn.ReLU() ) # FIX ME: Create the mean and logvar readout layers self.mu = nn.Linear(num_hidden, z_dims) self.logvar = nn.Linear(num_hidden, z_dims) # FIX ME: Create the decoder layers def decoder(self): self.decoder = nn.Sequential( nn.Linear(z_dims, num_hidden), nn.ReLU(), nn.Linear(num_hidden, num_hidden), nn.ReLU(), nn.Linear(num_hidden, input_size), nn.Sigmoid() ) def forward(self, x): # FIX ME: Implement the VAE forward function x = x.view(-1, self.input_size) h = self.encoder(x) mu = self.mu(h) Deep Learning 7. We will use induction. For the base case, t = 1, so we know that q(x1|x0) = ... inductive step, assume that ... We have q(xk+1|xk) = ... This gives q(xk|x0) and N(0, I) are independent Gaussian random variables, so q(xk|x0) is also a Gaussian random variable. We have E[q(xk|x0)] = ... 8. We have a = sqrt(r)Σ and Σ = (1 - ât)I. Since Σ is a constant multiple of the identity matrix, we have tr(Σ) = d(1 - ât) and det(Σ) = (1 - ât)d. Thus, we have DKL(q(xT|x0)||N(0, I)) = ... From equation 7, we have so ... Using equation 6, we get ... We know that ât = ... Deep Learning CNN with 4 convolutional layers and 3 MLP layers Learning Curve Epoch Train loss Train acc Val acc Deep Learning All Validation Accuracies Epoch MLP (depth 3) Validation Accuracy MLP (depth 7) Validation Accuracy CNN Validation Accuracy Laura Zhang 10/26/2025 STS.002 Professor Deringer Optimism vs. Elitism: Debating Control of the Stock Market From the late 18th century to the mid 20th century, finance evolved into a global system connecting both individuals and companies through credit and investment. The rise of industrialization, the expansion of railroads, and the World Wars created new demands for capital, making financial markets and personal wealth management. As these markets became an increasingly integral part of everyday life, debates over the role of finance in society also spread. With the emergence of powerful investment banks that influenced both the flow of capital and perceptions of stocks, people began to question who the stock market was truly benefiting. Finance & Society In 1928, investment counselor H. W. Moorhouse published What's Happening in Wall Street?, an article examining the bull market of the 1920s and its implications for America's economic future. Moorhouse argued that ordinary individual investors were the key driver of the stock market - their decisions, whether rational or irrational, played a central role in shaping market performance. He viewed the new era of finance in America as extensive participation by the public. Moorhouse believed that currently, individual investors made decisions without critical thought and often overlooked underlying causes of stock movements. He explains, "To whatever economic forces may have caused the "bull" market, Mr. Stebbins is grateful, even if he doesn't know what they are. He simply accepts what the ticker gives him. Unfortunately, very often what the ticker can give the market can also give away. If this sad event should come, Mr. Stebbins would again not know what has happened to him." For many investors, the stock market was a black box that produced prices at random. Because investors lacked a deeper 1. H.W. Moorhouse, "What's Happening in Wall Street?" The North American Review, December 1928, 673. Strava Case Executive Summary Strava is a social fitness platform that is considering whether to accept the term sheet offered by Sigma Partners. First, we calculated the NPV of Strava using the Pro Forma Financial Statements. We assumed that growth would be 0% after 2015, since growth rates are expected to continue decreasing in the future. We found a NPV of $122,238,371, so 39.19% of Strava's equity is $47,905,217.57, which is significantly more than the initial investment of $4,595,000. This implies that Sigma Partners would have to pay more for Strava's equity. We calculated the earnings for dividends for the Series A Preferred stock. We evaluated that the Series A Preferred shares are worth $1,148,750 more than a 39.19% stake in common equity would be. We performed sensitivity analysis and found that changes in growth rates and WACC can lead to large changes in NPV. For example, if the growth rates are 50% of projections and WACC is 20%, the value of the investment decreases to $24,212,769.71, which is significantly lower than before but still higher than the initial investment. Looking at other terms, we see that for liquidity preferences, investors could double their original investment before common shareholders receive more, which is less favorable for the founders. Managerial Finance Valuation of Strava To value Strava, we use the Pro Forma Financial Statements, which provide projected cash flows from 2011 to 2015. We assume that the financial statements predict accurate growth rates of users. We also assume that the percentage of paid users continues to be 20%. Beyond 2015, we predict a conservative growth rate of 0%, as we expect growth rates to continue declining over time. This is a perpetuity of $32,250,485 starting in 2016. We discounted the cash flows from 2011 to 2015 assuming a WACC of 16%, giving a present value of $26,270,398. We also discounted the terminal value, giving a present value of $95,967,973. Therefore, the NPV of Strava is $122,238,371. The value of 39.19% of Strava's equity is $47,905,217.57. This value is significantly more than the $4,595,000 purchase price. Valuation of Series A Preferred Shares If a common stock dividend is declared, the preferred stock must receive the dividend along with the common stock. This means that the difference between the preferred stock and common stock is the annual dividends for the preferred stock. We assume that the Board will issue dividends every year in the future and there is always sufficient cash flow available. These dividends are not cumulative, so preferred shareholders receive 4% of the $4,595,000 investment annually, which is $183,800. This is a perpetuity with a discount rate of 16%, so the difference between the preferred stock and common stock is $1,148,750. Waterhouse Investor Services Returns VW NYSE Equity Beta Debt % 0.153 0.043830 1.4024149 0.38 0.031 0.044050 Equity % (0.091) 0.037230 0.869497238 0.62 0.067 -0.020740 (0.219) -0.224870 (0.040) -0.072900 (0.125) 0.070410 (0.143) 0.045340 0.056 0.051610 0.263 -0.017060 (0.021) Managerial Finance 0.064 (0.200) 0.051390 (0.300) -0.007180 (0.071) -0.027940 0.231 0.037280 (0.156) 0.017650 0.037 -0.016400 (0.214) 0.020800 0.364 0.065940 -0.016360 0.267 0.021550 (0.105) 0.048530 0.176 0.039650 0.200 -0.004980 - 0.077720 (0.022) 0.022280 (0.022) -0.001750 (0.044) -0.029380 (0.070) 0.017850 (0.100) 0.018280